Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Proof. The first assertion about currents follows from the definition of minimal multiplicity and Siu decomposition. Homogeneity and subadditivity follow by scaling and adding testing currents. The subtraction identity is the corresponding property in the big cone, followed by a small Kähler perturbation. More explicitly, for \alpha_\varepsilon=\alpha+ \varepsilon\{\omega\}, subtract F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\} coefficientwise. In the big cone all positive currents contain this divisor, so subtracting it translates each minimal multiplicity by its coefficient. As \varepsilon\downarrow0, F_\varepsilon\to F. The class of F-F_\varepsilon can be absorbed in a Kähler perturbation tending to zero (there are only finitely many components). This gives the upper bound for the asserted identity; subadditivity applied after adding F gives the reverse bound. Weak compactness of positive currents gives pseudo-effectivity of the limit. This is also the subtraction property of the divisorial decomposition in [7, Section 3].

For the strict-transform formula, pull almost-positive testing currents to Y. Near the generic point of D', the map is an isomorphism, so the generic order is unchanged. This proves one inequality. Conversely, push a testing current on Y to X. To control its negative error, write the error as a small multiple of the fixed positive current \mu_*\omega_Y, and add a fixed smooth Kähler representative of C\omega_X-\{\mu_*\omega_Y\} for C sufficiently large. The result is a positive test in a Kähler perturbation tending to zero. The pushforward of \omega_Y has no divisorial order at the generic point of D, where \mu is an isomorphism. Thus this test has the same generic order as the original one at D'. Regularization, if needed, returns to tests with analytic singularities. This proves the other inequality and the formula. It is the analytic form of the strict-transform comparison in [7, Lemma 5.8].

Finally, choose analytic-singularity tests for a modified-nef class whose generic order at the chosen prime tends to zero. Subtract that generic divisorial order before restricting to a resolution of the prime. The remaining singular set does not contain its generic point, so restriction gives an almost-positive current there. The subtracted multiple and the Kähler error tend to zero. Taking the limit proves the last assertion; see also [7, Propositions 2.4 and 3.8].□

Proof. It suffices by Lemma 2.2 to prove E\leq N(\alpha). Write

with no common component, and suppose U\neq0. Its components are p-exceptional. Put n=\dim Y and \ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}. Normality gives \ell\leq n-2. Let \eta be the pullback of a Kähler form on X, let \omega be Kähler on Y, and set

The restriction property in Lemma 2.2 gives q(Z(\alpha),U)\geq0. The term q(p^*M,U) is zero: on each component of U, it contains \ell+1 factors pulled back from an image of dimension at most \ell. Distinct effective divisors have nonnegative intersection against the semipositive and Kähler factors in q. Hence

On the other hand, q(U,\eta)=0 by the same dimension count, whereas q(\eta,\eta)>0. The mixed Hodge index theorem, applied first with Kähler factors and then by a semipositive limit, says that q is negative semidefinite on \eta^\perp. It follows that q(U,U)=0 and that U is in the radical of q: the radical assertion follows from negative semidefiniteness on \eta^\perp, and then from q(U,\eta)=0. But

Indeed, a component with image dimension \ell has strictly positive integrand on a dense open. This is a contradiction. The mixed Hodge index statement used here is the mixed Hodge–Riemann relation [24, Theorems A and C]; the limit preserves the assertion that there is at most one positive direction.□

Proof. Subadditivity and nefness give N(\mu^*\alpha)\leq\mu^*R. By the strict-transform formula, the difference U=\mu^*R-N(\mu^*\alpha) is effective and exceptional. Moreover Z(\mu^*\alpha)=\mu^*\beta+\{U\} is modified nef. Apply Lemma 2.3 to this class. It gives U\leq N(Z(\mu^*\alpha))=0, proving the assertion.□

Proof. The divisor current [\operatorname{div}(s)]/m is a positive current in c_1(L), so Lemma 2.2 gives the order bound. In the stated decomposition, every section therefore divides holomorphically by the canonical section of mR. Conversely, multiplication gives a section of mL. The actual rational line identity makes these operations inverse after clearing denominators.□

Proof. Write X for the manifold. The assertion is immediate if \dim X=0, so assume \dim X>0. Every positive current T in c_1(L) contains [R]. The residual T-[R] is positive with zero cohomology class, so its mass against a Kähler form to the power \dim X-1 is zero. Thus it vanishes. For the second assertion, add [F] to any positive current in c_1(L-F). Uniqueness gives F\leq R, and subtraction in Lemma 2.2 gives the asserted negative part. A torsion line has zero Chern class and becomes trivial after taking a positive multiple, so the last statement is the same argument.□

Proof. With compatible canonical choices there is an actual rational divisor identity

Suppose q:V\to Y realizes the decomposition for the left side. Lemma 2.3, applied over X, gives

Cancelling q^*E_p in the actual rational line identities proves the first assertion. This argument also shows that further resolutions using the reduced-exceptional convention do not change the assertion \Ggood_n.

If J is nef, its pullback is nef, so its negative part is zero. The semiample line in the decomposition is therefore q^*p^*J itself, up to rational line isomorphism. Choose an actual Cartier multiple generated on V. Normality gives (p q)_*\OO_V=\OO_X, and projection formula identifies its sections with the sections downstairs. A base point downstairs would make every pullback section vanish on its nonempty fiber. There is no such point, so that multiple of J is globally generated.□

Proof. At x\in X, choose a holomorphic frame of L and put

The generic algebra A\otimes_RK is a product of finite field extensions of K, of total dimension d. The local meromorphic coefficient a of s belongs to this algebra. Since Z is irreducible and s is not identically zero, its restriction is nonzero on every generic local component. Thus a is nonzero in every field factor, and the determinant of multiplication by a is nonzero. Under a change of frame by a unit g, this determinant changes by g^{-d}. The determinants therefore glue to the asserted meromorphic section. The valuation formula for a norm over a discrete valuation ring gives the displayed divisor identity at every prime of the normal space X.

If s is holomorphic, then a\in A is integral over R. Its norm is integral over R and belongs to K, hence belongs to R by normality. This does not require A to be locally free. If s is nonzero at every point over x, then a is a unit in the finite semilocal algebra A; applying the same argument to a^{-1} shows that its norm is a unit in R.

For a meromorphic function f, use instead the determinant of T minus multiplication by its local coefficient. These monic polynomials agree on overlaps, and Cayley–Hamilton shows that they annihilate f. All these constructions use fractions of local analytic germs, so they remain available when the global meromorphic functions on X are constant.□

Proof. Replace Y by the smooth model in the hypothesis and pull F back. Write e^*L+F\sim_{\Q}R=N(e^*L+F). Pull back a positive current in c_1(L) and add [F]. Lemma 2.6 gives F\leq R. Put D_Y=R-F. Subtraction in Lemma 2.2 gives

In particular [D_Y] is the unique positive current in c_1(e^*L).

Choose m so that mL and mD_Y are integral and the latter is the divisor of a holomorphic section s of e^*(mL). Factor e through its normal Stein space:

The connected birational map h descends s to a holomorphic section s_Z of \nu^*(mL), by normality and projection formula. Lemma 2.8 gives a holomorphic section of mdL whose divisor is

Thus D=d^{-1}e_*D_Y is effective and satisfies L\sim_{\Q}D.

Both e^*D and D_Y are positive divisor currents in c_1(e^*L), so uniqueness gives e^*D=D_Y. We already know N(L)\leq D. If this inequality were strict at a prime A\subset X, choose analytic-singularity tests for L whose generic orders at A tend to \nu_A(L), and pull them to Y. For any prime A' dominating A, the generic orders are multiplied by \operatorname{mult}_{A'}e^*A. The pulled-back Kähler errors can be enlarged to Kähler errors upstairs tending to zero. Consequently

contrary to N(e^*L)=D_Y=e^*D. This proves D=N(L).□

Proof. The good-model assertion is the rational-boundary consequence of the full argument in [55, Proposition 2.5, Theorem 9.6, and Section 10]. That argument establishes the additional inductive and nonvanishing premises of its Proposition 2.5 before applying it, and proves a stronger real-boundary induction. Its only use of Assumption 1.1 is in its Lemma 6.1, on a resolved projective Albanese fibration with both boundaries zero. Thus its premise is precisely available here.

For clarity, its good-model conclusion gives on a common smooth projective resolution

as an actual rational divisor identity; the line on X_m is semiample. This is the comparison in [55, Lemma 2.2]. Algebraic nefness of a rational line on a projective variety implies analytic nefness, by adding arbitrarily small ample rational classes. Lemma 2.3 therefore identifies F with the analytic negative part of the left side. On a smooth projective variety, analytic pseudo-effectivity of a divisor class agrees with algebraic pseudo-effectivity [8, Proposition 1.2]; hence the good-model assertion applies to every projective instance of \Ggood_n. This proves the stated form of \Ggood_n. If the original adjoint is nef, the same comparison and normal descent give its semiampleness.□

Proof. We use Propositions 2.11 and 2.12, whose proofs occupy the rest of the paper. The assertion is immediate in dimension zero. Assume it in smaller dimensions and let (X,B) be as in Definition 2.1 in dimension n>0.

If a(X)=n, the Kähler Moishezon theorem makes X projective, and Proposition 2.10 applies. If 0<a(X)<n, the algebraic reduction of X is a dominant meromorphic map to a projective model of dimension a(X) [12, Theorem and Definition 3.1]. Proposition 2.11 applies.

It remains to consider a(X)=0. If X is simple, use Proposition 2.12. Otherwise Campana’s maximal covering-family theorem supplies a generically finite evaluation map from an incidence space which itself has a nontrivial fibration; see [13, Lemma 17 and Corollary 18]. After resolving the incidence, the graph, and the base, we obtain a smooth compact Kähler source X' and maps

where e is proper generically finite and Y' is in class \mathcal C [27, (1.1.1), (1.5)(3)–(4), (2.2)(1),(3), and (2.3)]. The algebraic dimension of X' is zero. Indeed, a meromorphic function on X' descends through the birational part of the normal Stein factorization of e. Lemma 2.8 gives its characteristic polynomial over the finite part, with meromorphic coefficients on X. These coefficients are constant because a(X)=0, so the function is constant on the irreducible space X'. Bimeromorphic modifications do not change this conclusion.

Choose a reduced SNC divisor B' containing the inverse image of \Supp B, after a further resolution. Pullback of logarithmic differentials gives the actual rational identity

For example, this follows locally by pulling back logarithmic top forms for the reduced support of B; decreasing its coefficients to those of B only increases the error. Thus the adjoint upstairs is pseudo-effective. Proposition 2.11 applies to (X',B'). Its semiample positive part is torsion, since a nonconstant semiample map would give a nonconstant meromorphic function on X'. Proposition 2.9 now gives the required decomposition for K_X+B. This exhausts the cases and proves the induction.□

Proof of Theorem 1.2. Apply Theorem 2.13 on a log resolution of the given lc pair, with the reduced-exceptional boundary. The adjoint is pseudo-effective because the original one is nef. Proposition 2.7 then gives generation of an actual Cartier multiple at every point of the original normal space.□

Proof. These are the semiampleness, proper relative model and finite-model assertions of Theorem 3.55, proved below by dimension induction independently of \Ggood. Finite marked weak-model geography is recorded in Lemma 3.54. Projective analytic model constructions used in that induction are supplied by Proposition 3.25. None of these statements promotes an undetected generalized-ray contraction to a projective map.□

Proof. For dlt input, decrease the floor by a sufficiently small positive rational multiple. The resulting pair is klt and remains f-antiample. For this klt pair the base point free hypothesis holds for L, because L has zero fiber degrees and the negative adjoint is relatively ample. Over a smaller base neighborhood every sufficiently high power of L is therefore generated. The associated morphism is constant on each connected fiber: its tautological line has degree zero on every fiber curve, and every positive-dimensional projective image contains a curve. Its graph then factors through Z; the graph projection is finite bimeromorphic and Z is normal. Descend two consecutive generated powers L^k,L^{k+1} to lines M_k,M_{k+1}. The line M_{k+1}\otimes M_k^{-1} pulls back to L. Projection formula identifies this quotient with f_*L, so these local descents and their evaluation maps agree on overlaps. Pull the descended line to the positive side of a flip. This is the argument of [54, Lemma 7.4].□

Proof. We give the reduction to the local finite-generation input, including why it yields a single finite program over the compact base. Let \mathcal N_{\R} be the finite-dimensional space of degrees of global rational lines on curves over V, and put r=\dim\mathcal N_{\R}. If r=0, then J is already curve-nef. Assume r>0. Choose rational lines H_1,\ldots,H_r whose degree classes form a basis and such that both H_i and J+H_i are relatively ample. Such a basis exists because the relatively ample classes whose sum with J is also relatively ample form a nonempty open set containing sufficiently positive classes. Define the rational simplex of adjoints

We will choose a general direction H=\sum_i\theta_iH_i with \theta_i>0 and \sum_i\theta_i=1. Its entire scaling segment lies in this simplex:

and it lies in the relative interior for 0<t<1.

We produce the klt representatives for these exact vertices on each sufficiently small Stein neighborhood separately. A relative Cartier line and its inverse have local effective representatives there: their proper direct images are coherent of generic rank one, so Cartan generation supplies nonzero local sections. This uses that \pi is bimeromorphic. Represent each H_i by a divided general free effective divisor G_i\sim_{\Q}H_i, using a sufficiently divisible relatively generated multiple. These finitely many general divisors preserve klt with B. Choose a relatively ample integral line A, and local effective representatives A^+\sim A, A^-\sim -A by the preceding argument. For one sufficiently small positive rational \epsilon, all boundaries

are effective and klt. Their adjoints represent J and J+H_i, respectively, and \epsilon A^+ is a common effective relatively ample summand. In particular, every rational adjoint in \Pi is relatively big.

On this Stein member, choose one simultaneous projective log resolution of the finite family. Give each exceptional prime a coefficient in (\max\{0,c_0,\ldots,c_r\},1), where the c_j are its crepant coefficients for the vertex pairs. If E\geq0 is resolution-exceptional with -E resolution-ample, then subtracting a sufficiently small multiple of E from the pullback of \epsilon A^+ leaves a relatively ample rational divisor. A sufficiently small positive rational multiple of this divisor can be removed from every boundary while leaving all coefficients effective and below one; the strict exceptional margins ensure this along the exceptional primes. A divided general free representative of this common summand preserves the simultaneous SNC condition. The smooth multigraded theorem now applies to these vertices. Exceptional corrections leave the local adjoint rings unchanged by projection to a normal space; clearing the finitely many line identifications simultaneously makes them multiplicative. This is the reduction in [54, Section 7.2], using [18, Theorem 3.1].

Here are the two consequences of finite generation that we need. The normalized main relative Proj of any rational adjoint \Theta\in\Pi extracts no prime divisor, and its trace is a relatively ample rational line. Indeed, resolve the degree-one base ideal after a common Veronese and write

with H_\Theta tautological and relatively ample. Generation says that all sections in degree k vanish at least along kG. If a component of G were not q-exceptional, relative generation of q_*\OO(G)\otimes H_\Theta^k would supply a section with smaller vanishing there. If a p-exceptional prime Q were not q-exceptional, the same argument with q_*\OO(Q)\otimes H_\Theta^k would contradict p_*\OO(Q)=\OO_T. This proves nonextraction. There are also only finitely many marked normalized main Proj models, where the marking records their common bimeromorphic open over V. To see this, choose finitely many homogeneous generators locally. Proj charts of each diagonal ring use homogeneous monomials as denominators; the supports of those monomials in the finite set of generators determine the degree-zero localizations and their gluing. Only finitely many support patterns occur. Normalization and taking the main component preserve this finiteness. A finite cover of the compact base gives finitely many global marked models, since marked local identifications agree on the common dense open and glue uniquely. These are [54, Lemmas 7.5 and 7.6].

For completeness, we construct the scaling program controlled by this finite list. On a working model, let A_0 be a relatively ample line. The closed curve cone in the dual of its global degree space has compact slice A_0\cdot z=1: for every line D, both kA_0+D and kA_0-D are relatively ample for k large, which bounds every coordinate. Local cone truncations on the finite Stein cover express the negative part of this slice, after any positive ample truncation, using finitely many actual curve classes. A negative extremal ray can therefore be separated by a rational nef support annihilating just that ray. A positive multiple of the support minus the klt adjoint is relatively ample. Relative base point freeness contracts exactly the ray. If the contraction is divisorial, its single exceptional prime and Lemma 3.3 prove the global strong property downstairs. If it is small, the relative Proj of the adjoint is its small positive model. For any global line D on the negative model, killing its ray degree and applying the same lemma gives an actual rational identity

It transforms to the positive side, proving the global strong property there. This is the continuation construction of [54, Proposition 7.7]; it also preserves projectivity over V and compact Kählerness.

The transforms of H_1,\ldots,H_r continue to span the degree spaces: use pullback in a divisorial step and the preceding decomposition of each line in a small step. A line of zero global relative degrees stays so by Lemma 3.3; curves in the contraction bases can be lifted through projective morphisms. There are only countably many possible finite sequences of curve rays, and each determines its contractions and flips uniquely. We may therefore choose H in the relative interior of \operatorname{conv}\{H_1,\ldots,H_r\} so that it avoids all ties between independent ray degrees in advance. At a positive nef wall, write w=J+tH for the traces on the current model, and choose a preceding parameter s>t for which J+sH is relatively ample. On the zero face of the normalized compact curve slice,

in degrees. Thus J is uniformly negative on that face. Choose the positive ample truncation so that its compact remainder K misses the face. The finite rational polyhedral test and the general choice of H make the face a single ray R. The wall class has a positive minimum on K and is positive on every other truncated curve generator. Hence a sufficiently small open neighborhood of w inside the annihilator R^\perp consists of nef classes with zero face exactly R.

Suppose first that R^\perp has positive dimension. It is a rational hyperplane because R is represented by an integral curve. Choose a rational simplex of support classes in that neighborhood containing the degree class of w in its relative interior, and represent each vertex by a rational line N. For each such N, compactness gives an integer a>0 for which aN-J is relatively ample. Relative base point freeness contracts exactly R and descends N to a relatively ample rational line on the contraction base. These bases agree because the contracted curves agree. The difference between w and the corresponding convex combination of these rational lines has zero relative degree. In the finite rational span of the lines involved, the degree map has rational kernel, since rational lines have rational degrees on integral curves. The difference is therefore a real combination of rational lines of zero relative degree. Lemma 3.3 descends these lines, and lifting curves from the base shows that their descents still have zero degree over V. They do not affect relative ampleness. Convexity now shows that w descends to a relatively ample real line class over V. If R^\perp=0, the same zero-degree descent applies to w; the contraction base has no curves over V, hence is finite over V, and relative ampleness is automatic.

After a flip, let w^+ be the pullback of this wall class to the positive side. For sufficiently small \delta>0,

is relatively ample over V: the wall is pulled back from a relatively ample class on the contraction base, and J^+ is relatively ample over that base. After a divisorial contraction, openness of the ample cone on the base gives the same nonempty interval of relative ampleness.

Each working model is consequently one of the marked ample models already counted. Indeed, for a fixed finite prefix and an interior parameter 0<t<1, replace the direction and parameter by nearby rational ones. They remain in \Pi, and all strict signs in that prefix and the final relative ampleness persist; the associated relative section ring has this working model as its Proj. A repeated marked working model would have the same trace of B, because all steps are nonextracting, and hence the same discrepancies. This contradicts the strict increase in Equation (3.1) at an intervening nontrivial step. The finite list therefore forces termination. The last adjoint is curve-nef, since the continuation construction applies whenever it is not. Composing Equation (3.1) proves the final comparison.□

Proof. Choose a small rational \epsilon>0 for which (S,\Delta-\epsilon D) is effective and klt. On a projective log resolution r:W\to S, use the strict lowered boundary and give every exceptional prime a coefficient strictly between its crepant coefficient and 1, and at least zero. The resulting klt adjoint has the actual form

where every r-exceptional prime has positive coefficient in F. Apply Proposition 3.4 over S. On its endpoint the trace F_Y is effective, exceptional over S, and relatively nef. Negativity makes F_Y=0. Since all initially exceptional primes had positive coefficient, Y\to S is small and the lowered adjoint is crepant. Restoring \epsilon D gives the full crepant identity.

A projective small morphism to a smooth germ is an isomorphism: transport a relatively ample line to the smooth germ, where it is Cartier; smallness makes the original line its pullback, contradicting relative ampleness on a nontrivial fiber. Thus h is an isomorphism over an SNC open meeting all lc centers of the dlt pair. The full crepant comparison then proves dlt on Y.□

Proof. Write H=J in the first case and H=J+bP in the second. Fix \omega with \{H\}+\omega Kähler. For each 0<\eta<1, choose a positive rational \epsilon so small that, with F=\lfloor B\rfloor,

For \lambda\in[\eta,1] use the presentations

omitting b\{P\} in the first case. Their b-data are globally nef on the fixed initial carrier, and their boundary-plus-nef traces are globally modified big. These are effective gklt data.

We first construct a step whenever the unscaled working trace is not nef. Choose a positive shift strictly below the current wall and use the gklt presentation of its current trace A. The ray at the wall is A-negative. On a compact slice of the analytic curve cone, the cone theorem makes the A-negative part locally polyhedral after a sufficiently small positive Kähler truncation. Separating the chosen extremal ray from the other finitely many rays in that truncation gives a Kähler class \omega_R such that A+\omega_R is nef and its zero face is precisely this ray; see the supporting construction in Proposition 3.48. Add the pullback of this new Kähler class to the fixed nef carrier. The resulting adjoint still has globally nef data and modified-big boundary-plus-nef trace, so the nef assertion of Proposition 3.2 supplies its supporting contraction. This uses a new supporting direction on the current space; it does not require the trace of the original scaling direction to be Kähler. We verify a rational ordinary detector before asserting projectivity or constructing a flip. In the second case nefness of P makes J negative on the ray; the first case has this property directly. Lower the floor by a small rational amount, retaining negativity of J_\epsilon, so that -J\cdot R\leq2(-J_\epsilon\cdot R). The ordinary klt cone length bound gives a rational curve generator C with

In the second case, if P\cdot C>0, integrality of mP gives P\cdot C\geq1/m, contradicting

Thus in that case the ray is P-trivial. The lowered ordinary adjoint is a global rational detector. The detected projectivity and flip constructions in Lemma 3.16 and Proposition 3.18 supply the projective ordinary step. The difference of J and a positive rational multiple of the lowered adjoint has zero contracted degree; exact descent in Lemma 3.3 shows that restoring the floor preserves the positive-side sign. This is the ordinary replacement of Proposition 3.19, and the full pair remains dlt by Equation (3.1).

In the second case, Lemma 3.3 descends the actual Cartier line mP without changing m. Analytic nefness descends and pulls back under these projective contractions [39, Lemma 2.44]. A generated multiple descends as a generated line by projection formula and pulls back as one. Thus the same length estimate and the same b apply at the next step. The corresponding assertion on a final Mori ray follows from exactly the same estimate. For a semiample P, the map of its fixed generated multiple is constant on each connected projective contracted fiber: otherwise its image contains a curve of positive degree. The map therefore factors through the normal contraction base, including in the fiber-type case.

The traces used in this construction are linear by the degree-zero descent description preceding Lemma 3.3. At a wall the scaled class is pulled back from its base. The comparison for an earlier parameter is a nonnegative multiple of the ordinary adjoint comparison. Consequently the gklt presentations on any fixed positive segment remain gklt along its program, and every working model is a weak model for a member of that segment. The finite marked weak-model assertion in Proposition 3.2 applies. A marked model cannot recur: its boundary trace would be the same, whereas a nontrivial intervening ordinary step strictly increases a discrepancy. There are therefore only finitely many steps with threshold at least \eta.

For decreasing cutoffs, any already-constructed finite prefix remains valid. The floor perturbations cancel in the displayed classes and continue to cancel under their linear transforms; their nonpositive comparisons preserve the new gklt data. Starting at its last nef wall therefore extends the prefix to the smaller cutoff. This gives one compatible scaling program. A positive limiting threshold would lie in a finite positive truncation, so an infinite program has limit zero. Pseudo-effectivity excludes a negative Mori endpoint. If H is not pseudo-effective, its pseudo-effective threshold in the initial Kähler direction is positive. Every nef working trace, together with its effective exceptional comparison, makes the initial scaled class pseudo-effective, so the scaling thresholds are bounded below by that positive number. The program is therefore finite. Its endpoint cannot be nef, since that comparison would make H pseudo-effective; it is the stated Mori fiber contraction.□

Proof. For \lambda in the working interval of T_i, every earlier threshold is at least \lambda. Each earlier step is therefore negative or trivial for this parameter. Composing its comparisons gives Equation (3.2). Since \alpha_i(\lambda) is nef, Lemma 2.3 identifies the exceptional divisor with the negative part.

On the fixed resolution U, choose a Kähler class \Omega and c>0 such that c\Omega-r^*A is Kähler. Monotonicity of each minimal multiplicity under addition of a nef class gives

The left side tends to the right side by the definition using small Kähler perturbations. This proves the convergence on U, and the same argument applies on a resolution representing any chosen divisorial place.□

Proof. Lemma 2.4 makes the stated equality valid on any higher resolution. Choose b as in Lemma 3.6. Adding bP leaves the same negative part. Indeed, on a smooth resolution write \alpha=\pi^*\{J\}, p=\pi^*\{P\}, and e=\pi^*E=N(\alpha). Subadditivity and homogeneity of negative multiplicities give

Thus N(\alpha+bp)=e.

Run the scaling for H=J+bP. Let Q be the strict transform on a fixed resolution of a prime component of E. Its multiplicity in N(\pi^*\{H\}) is positive. The limit in Lemma 3.7 makes its multiplicity positive for every sufficiently small positive parameter. If that prime still survived on the corresponding working model, Equation (3.2) would make its multiplicity zero, because the comparison there is exceptional over that model. It must therefore be contracted after finitely many steps. The same conclusion holds if a nef endpoint is reached before taking the limit. There are only finitely many components of E, so after a finite prefix all have disappeared.

Throughout the program P descends and remains nef. The actual identity J_i\sim_{\Q}P_i+E_i, with E_i the codimension-one pushforward of E, follows from the reflexive trace convention. Once E_i=0, we have J_i\sim_{\Q}P_i; since P_i is nef, the program stops. Conversely, a divisorial negative step can contract only a prime of E_i: its ray is P_i-trivial, so the effective divisor E_i has negative degree on every contracted curve, and those curves are contained in its support. This proves the exact assertion about contracted primes.□

Proof. For a modification of a smooth compact Kähler manifold, the degree-two modification theorem gives these decompositions, and the exceptional classes have type (1,1). Suppose they hold at one step. Its contraction base and both working spaces have rational singularities: slight lowering of the rational Cartier floor makes the ordinary pair klt, and the detected contraction has a base with rational singularities. These contractions are bimeromorphic, so a common resolution and Leray give R^if_*\OO=0 for i>0; this verifies the additional hypothesis in Lemma 3.15.

The ray of this detected contraction is one ray for Bott–Chern degrees. This also suffices for degree-two classes. Indeed, pull a class \gamma\in H^2(V,\R) to a smooth resolution p:U\to V and take its Hodge decomposition. By the decomposition already proved for this working model in Equation (3.4), its (1,1)-part is p^*\beta+\{E\} for a Bott–Chern class \beta and an exceptional real divisor E. The other Hodge components have zero degree on curves. On every p-contracted curve the degree of E is therefore zero, so both signs of negativity make E=0. Lifting any curve of V through the projective resolution now shows that \gamma and \beta have the same curve degrees. Thus subtracting a multiple of \{J\} from either a degree-two or a Bott–Chern class can annihilate every curve of the contracted fiber. By Lemma 3.15, a degree-two class, or a Bott–Chern class, with these zero degrees is pulled back from the base. Subtract a multiple of \{J\} to make any given class annihilate that ray, descend the remainder, and use the trace \{J^+\} to define its transform. On a common resolution the difference is a multiple of the exceptional adjoint comparison in Equation (3.1). For a flip the two exceptional prime sets on that resolution are the same. For a divisorial contraction the target resolution has just the one additional exceptional prime of the step. The old decomposition therefore spans the new one and proves surjectivity of the transform.

The sums are direct. If the pullback of a class from V is an exceptional divisor class, that divisor has zero degree on all curves over V. Applying exceptional negativity to both signs makes the divisor zero; injectivity of pullback gives that the class is zero. The pullback injectivity in degree two and in Bott–Chern cohomology is also Lemma 3.15, applied to the resolution. Passing to higher smooth resolutions proves the assertion for every p.

For the line assertion, let \mathscr H be a holomorphic line on U, and put \mathscr Q=(p_*\mathscr H)^{**}. Global strong \Q-factoriality gives an invertible sheaf \mathscr M=\mathscr Q^{[m]} for some m>0. The coherent evaluation comparison of [54, Lemma 7.2] gives

Indeed (p_*\mathscr H^{\otimes m})^{**}=\mathscr M, since the two sheaves agree at the general points of all divisors of V. Local generators of p_*\mathscr H^{\otimes m} in a frame of \mathscr M compare their evaluations with the pulled-back coefficients by meromorphic quotients. Those quotients agree on the isomorphism locus and hence glue; their zeros and poles are exceptional. Taking Chern classes proves the assertion without choosing a meromorphic section of \mathscr H.□

Proof. A pluricanonical section makes K_X pseudo-effective. By \Ggood_d, on a smooth modification \mu:\widetilde X\to X there is an actual identity

Lemma 2.5 identifies the divisible section spaces with those of P. Hence \kappa(P)=0; a semiample line of Iitaka dimension zero has a trivial positive multiple. Proposition 2.9, applied with e=\mu and F=0, gives an effective rational divisor E_X=N(K_X) with K_X\sim_{\Q}E_X. Lemma 2.4 then gives N(\mu^*\{K_X\})=\mu^*E_X. Apply Proposition 3.8 with P=0. It reaches a model with torsion canonical line. It is terminal: exceptional places over the initial smooth space have log discrepancy greater than 1, and a divisor contracted from that space starts with log discrepancy 1 and acquires a strict increase in Equation (3.1). Finally apply Lemma 3.9.□

Proof. Resolve the dlt pair, giving new exceptional primes coefficient one. The resolved adjoint is the pullback of J plus an effective exceptional divisor. Apply \Ggood_d to this smooth pair. Lemma 2.3 subtracts the exceptional resolution error from its negative part. On a smooth higher model r:\widetilde S\to S we obtain the actual identity

Every component of E is exceptional over S and has center disjoint from the floor, by Equation (3.5). Choose m divisible enough that mJ,mP,mE are integral and mP is generated. Lemma 2.5 shows that multiplication by the section of mE identifies the complete section spaces. Over a neighborhood of the floor, E is absent upstairs. The line mr^*J is generated there, and these sections descend to S by normality. Thus mJ is generated on a neighborhood U of the entire floor.

Let \rho_\Gamma:\Gamma\to S be the normalized graph of the map defined by the complete system |mJ|, and write \varphi:\Gamma\to\PP(H^0(S,mJ)^*) for its morphism to projective space. The map \rho_\Gamma is projective and is an isomorphism over U. The rational line

is semiample. The generated moving system |mP| induces a morphism \psi:\widetilde S\to\Gamma whose pullback of P_\Gamma is P. Choose a small positive rational \epsilon for which (S,\Delta-\epsilon D) is effective and klt. On a projective log resolution W_0\to\Gamma\to S, give every exceptional prime a nonnegative coefficient strictly above its crepant coefficient for the lowered pair and strictly below 1. Its klt adjoint equals the pullback of J-\epsilon D plus an effective error positive on every prime exceptional over S.

Run Proposition 3.4 over \Gamma, and write g:Y\to\Gamma for the endpoint morphism and h=\rho_\Gamma g:Y\to S. The surviving error F_Y is effective and has positive coefficient on every prime of Y exceptional over S. Over U, where \Gamma=S, it is also exceptional and relatively nef, since the lowered adjoint is relatively nef and its other term is pulled back from S. Local exceptional negativity makes it zero there. Hence h is small over U, and it is an isomorphism over the smooth part of U by the argument in Corollary 3.5. Restore the floor. The full pair on Y is dlt near it, by the crepant small comparison and the SNC open, and is klt elsewhere. No exceptional prime has center meeting the floor, so h^*D is its strict transform and

Put P_Y=g^*P_\Gamma. On a common smooth model a:\widehat S\to\widetilde S, b:\widehat S\to Y over S, the maps \psi a and g b agree on the common dense open and hence everywhere by separatedness of \Gamma. Thus a^*P\sim_{\Q}b^*P_Y. Define E_Y^0=b_*a^*E. This is an effective rational divisor, exceptional over S and supported away from the floor. Pushing Equation (3.6) to Y gives

The global strong property on Y makes E_Y^0 rational Cartier. The divisor a^*E-b^*E_Y^0 is b-exceptional and rationally linearly trivial. Exceptional negativity applied to both signs therefore gives a^*E=b^*E_Y^0.

Define E_Y=E_Y^0+F_Y. Adding the restored-boundary error gives

where E_Y is exceptional over S, is supported away from the floor, and contains every prime exceptional for h. It is rational Cartier by the global strong property. On \widehat S, Lemma 2.4 gives

Exceptional translation for b^*F_Y over S then gives N(b^*\{J_Y\})=b^*E_Y. The hypotheses of Proposition 3.8 are now satisfied. Each contracted curve lies in \Supp E_Y, since its P_Y-degree is zero and its E_Y-degree is negative. Nontrivial connected projective fibers are covered by curves, so this contraction and its positive replacement are unchanged over a neighborhood of the floor. The same assertion persists with the pushed-forward effective divisor at each step. All primes exceptional over S are contracted and no prime coming from S is contracted. The endpoint M is therefore small with S, has the asserted morphism over U, and has J_M\sim_{\Q}P_M semiample.

Pull \gamma from S to Y. Although Y may be singular, its subsequent program consists of the detected ordinary analytic-ray steps in Proposition 3.8. At step i, choose c_i\in\R so that \gamma_i-c_i\{J_i\} has degree zero on its contracted ray. All curves in a fiber have class on that same analytic ray. The working spaces have rational singularities after slightly lowering the Cartier floor, and relative vanishing gives rational singularities on the contraction base. Lemma 3.15 therefore descends this class uniquely to the base. Pull it to the positive side and add c_i\{J_{i+1}\}, defining \gamma_{i+1}. The pullback difference is c_i times the ordinary adjoint comparison, hence is an actual real exceptional divisor class. No smooth-start cohomology splitting is needed for this construction.

Telescope these comparisons on a common resolution. The original map Y\to S was a morphism, and \gamma was pulled back through it. Since S and M match in codimension one, the resulting divisor is exceptional over both. Each step is an isomorphism over the chosen neighborhood U of the floor. There the class remains the pullback from S; the corresponding comparison divisor has zero class and is exceptional. Negativity applied to both signs makes it zero there. Thus its support is disjoint from the inverse image of U, as claimed.□

Proof. Let P_M=J_M denote the semiample rational line from Lemma 3.11, and use its transform A_M. On a common resolution p:W\to S,\ q:W\to M, let F_A be the real divisor in the common exceptional span defined by

It is effective: -F_A is q-nef, because p^*A is nef and the other term is pulled back from M; apply exceptional negativity. Its support is disjoint from p^{-1}(U). In particular A_M is pseudo-effective, and its negative multiplicity at every prime of M is zero. Indeed, q^*A_M=p^*A+\{F_A\}, and Lemma 2.3 applied over S gives N(q^*A_M)=F_A.

Let D_M be the rational Cartier trace of D. Choose \tau>0 so small that A+\tau\{D\} is Kähler. For a fixed sufficiently small t>0, consider on M the boundary \Delta_M-t\tau D_M and the nef b-class represented on W by t p^*(A+\tau\{D\}). Its trace on M is t(A_M+\tau\{D_M\}), so its generalized adjoint is

These are effective gklt data. Over U the b-class descends to M, so its generalized discrepancy contribution is zero there, and subtracting t\tau D_M raises every zero log discrepancy of the dlt pair. Away from the floor the ordinary pair is klt; on one fixed log resolution its finitely many subunit coefficients stay subunit for small t. This proves gklt. The generalized boundary is big. On a common resolution the pullback of the trace of A+\tau\{D\} minus p^*(A+\tau\{D\}) is effective exceptional by the same negativity argument as for F_A. The latter pullback is nef and big, so the trace is big; the remaining boundary is effective.

Choose an integer m>0 such that mP_M is generated, and let g:M\to Z be the Stein factor of its morphism. Then Z is normal projective and mP_M=g^*H for an ample generated Cartier line H on Z. Set L=\{P_M\}+tA_M. The preceding preparation makes L an effective gklt adjoint with globally nef b-data and globally modified-big boundary-plus-nef trace. It is pseudo-effective because P_M is nef and A_M is modified nef. Apply the proper relative assertion of Proposition 3.2 over Z. This produces a chosen nonextracting good log terminal model \phi:M\dashrightarrow V and a morphism g_V:V\to Z, with L_V nef over Z. The hypothesis here is properness: the semiample morphism M\to Z need not be projective. Put

The relative construction preserves the actual line pulled back from Z, and its forward Bott–Chern transport preserves the displayed linear relation; these are the transforms of P_M,A_M.

The map \phi is small. On a common projective resolution a:W'\to M, c:W'\to V, its comparison is

where E is exceptional over V and has positive coefficient at the strict transform of each prime contracted from M. The forward trace L_V is pseudo-effective, so exceptional translation gives

But L is modified nef on M; the coefficient of N(a^*L) at a strict prime of M is zero. Thus no prime of M is contracted. Nonextraction then proves smallness.

Fix b>2dm. We claim that L_V+b\{P_V\} is absolutely nef. Otherwise the cone theorem for the effective gklt adjoint L_V gives an L_V-negative rational extremal generator C such that

Indeed the cone summand on which L_V is nonnegative also has nonnegative P_V-degree, so a negative ray summand must account for any failure of nefness. If P_V\cdot C=0, ampleness of H makes C vertical over Z, contradicting relative nefness of L_V. Otherwise Cartier integrality gives P_V\cdot C\geq1/m, contradicting

This proves the claim. Both \{P_V\} and (1+b)\{P_V\}+tA_V are nef. By convexity,

Thus \delta=t/(1+b) works. The actual line P_V has the same generated multiple as P_M, since both are pulled back from H/m. Its class is the transform of J. The relative comparison and linear transport give exceptional-divisor comparisons for P_M,A_M. Since S,M,V agree in codimension one, their final comparison divisors are exceptional over both S and V, as required.□

Proof. We first choose a scaling with strict positive walls and Kähler interior classes. We then reduce the transformations on each log canonical stratum to small diagrams away from its smaller strata. Finally, the single nef interval supplied above rules out infinitely many such diagrams.

Choice of scaling. For every possible finite prefix of the detected ordinary truncation scalings from X, Lemma 3.9 makes the transform of H^{1,1}(X,\R) onto the working Bott–Chern space surjective. There are countably many such sequences: curve rays belong to countably many integral homology classes, and a ray determines its contraction and its flip uniquely. On each working space, two distinct curve rays C,C' with nonzero L-degrees give a proper hyperplane of initial directions defined by

where the degrees use the traces on that working space. Whenever both rays define finite walls, this equation says that their wall parameters agree. Avoid all these hyperplanes, and also avoid \omega\cdot C=0 whenever L\cdot C=0. A general Kähler direction, enlarged to make the initial scaled class Kähler, has these properties.

At a wall the scaled class is pulled back from the contraction base. Immediately after a flip, curves in an opposite-side fiber have positive L-degree. Another negative ray at the same wall would satisfy the excluded wall equation with that opposite-side ray. Immediately after a divisorial contraction, lift any putative new wall curve through the projective morphism. Its lifted ray is distinct from the contracted ray and satisfies the same wall equation; if its L-degree were zero, its \omega-degree would also be zero. Both possibilities were excluded. Hence consecutive positive thresholds strictly decrease, and each working interval has nonempty interior. Its interior nef class is big: the class on X is a positive Kähler perturbation of the pseudo-effective class L, and bigness is preserved under the birational comparison. No rational curve can have zero interior degree. An L-negative or L-positive such curve would violate nefness at one of the two nearby parameters; the zero-L case was excluded. The gklt presentation of a positive truncation and [38, Theorem 4.3] therefore make the interior class Kähler. By Lemma 3.6, infinitely many thresholds would tend to zero.

Reduction to small transformations on a stratum. Suppose the sequence is infinite. The homological divisor count [54, Lemma 7.8] discards all divisorial ambient steps after a finite prefix. We next reduce the remaining flips to small transformations on a stratum. For a working ambient pair (X_i,B_i), ordinary dlt adjunction on the normalization S of an lc center gives an effective dlt pair (S,\Delta_S) and the actual rational-line residue identity

Its lc centers are the proper nested lc centers. The coefficients of these differents form a DCC set depending only on the original coefficients and the chain length. At one restriction they have the form

which preserves DCC under iteration. These assertions, including the actual residue comparisons, are [54, Lemma 7.16]. Restricting Equation (3.1) along a strict adjunction chain gives an effective comparison on strata, strictly positive for a place whose center maps into the ambient exceptional locus [54, Lemma 7.17]. Both lemmas are dimension-free.

Identify a surviving lc center with its strict transform on the common isomorphism open. There are finitely many lc centers at each stage, and their surviving list stabilizes. A new zero-discrepancy place was already a zero-discrepancy place, and strictness keeps the general point of its center outside the surgery. Starting with the smallest centers, induct on their dimensions. Fix a surviving center and suppose the loci already avoid all its proper subcenters. We will prove that the loci eventually avoid this center as well. The induced diagrams on its normalized strata are then isomorphisms near their non-klt loci. On each stratum, let D_S be the sum of the restrictions of the ambient floor components not used in its generic adjunction chain. Strong \Q-factoriality of the ambient space makes D_S rational Cartier, and the nested-center description gives \Supp D_S=\Supp\lfloor\Delta_S\rfloor. Slightly lowering this effective divisor makes the stratum pair klt.

After a further tail the induced transformations on this stratum are small and their boundary transforms agree. Indeed, a prime extracted in a restricted transformation has discrepancy strictly less than 1 before extraction: its new boundary is effective and strict comparison increases its discrepancy. Its center is away from the unchanged non-klt locus. At the start of this tail there are only finitely many such places. On a fixed compact log resolution, the positive SNC weights 1-b_j have a positive minimum, and the Jacobian inequality bounds every place below the strict cutoff 1; include also the finitely many nonexceptional positive-boundary primes. This is [54, Lemma 7.9], with its lc assertion away from the non-klt locus. Discrepancies do not decrease, so all later extractions belong to this same finite list. Repeated extraction of one place would give a strictly decreasing sequence of its boundary coefficients, contrary to the adjunction DCC property.

Once extractions stop, [54, Lemma 7.8] makes divisorial contractions finite. Its dimension-free count is the dimension of the span of prime divisor cycles in H_{2d-2} of the compact d-dimensional stratum. Restriction to the common isomorphism open uses Borel–Moore homology: removing codimension at least two from the target has no effect in this degree, while a lost prime has nonzero class by its positive Kähler volume. The count strictly drops. The coefficients on the finitely many matched boundary primes then stabilize by monotonicity and DCC. Neither morphism of a restricted diagram can still contract a divisor to the common intermediate space, since strict comparison would change the discrepancy of that matched divisor. The diagrams are therefore small, with identical boundary transforms.

We justify the passage from an isomorphic restricted diagram to absence of ambient surgery near the stratum, also in higher codimension. Its identified boundaries give the same adjunction data, not only isomorphic underlying spaces. Choose a common projective log resolution preserving the general points of the surviving lc centers, and restrict its adjoint comparisons along the strict adjunction chain described above. The generic point of each surviving center lies outside the surgery: a zero-discrepancy place remains such a place, whereas a center contained in the surgery would acquire strictly larger discrepancy. Thus the strict transform S_W of the normalized stratum is not contained in the comparison support. For a zero-dimensional stratum this already excludes any intersection with the surgery, so suppose its dimension is positive. Its restriction is consequently a well-defined effective divisor. Strict adjunction identifies its class with the difference of the two pulled-back stratum adjoints. When the restricted diagram is an isomorphism with the same boundary, that difference is zero. An effective nonzero divisor on the compact Kähler resolution of S_W has positive mass against a Kähler power, so the restricted divisor is zero.

If an ambient step nevertheless met the stratum, take the first such step. The full-support assertion following Equation (3.1), and surjectivity of S_W onto the stratum, force its comparison support to meet S_W. Noncontainment makes this a nonzero effective restriction, and the cumulative comparison dominates it. This contradicts the vanishing just proved. Earlier steps are isomorphisms on a neighborhood of the stratum by the choice of this first step. The ambient diagram is therefore an isomorphism near the whole stratum. This verifies the support conclusion used in the induction on proper subcenters; for a higher-codimension stratum it is noncontainment in the support, not merely being different from a divisor component, that is needed.

Excluding the remaining walls. It remains to show that only finitely many of these small diagrams are nontrivial. Write S for the first stratum in this last tail and J=K_S+\Delta for its adjoint. Let its restricted original scaling be \{J\}+\lambda H. Choose an interior parameter \lambda_0>0 on this working space, above the remaining walls, and set

It is Kähler, as the restriction of the interior ambient Kähler class. The exact change of parameter is

It applies for 0\leq\lambda<\lambda_0. The new walls t_i strictly decrease toward zero, and the traces \{J_i\}+tA_i are nef on their successive restricted working intervals.

On a common resolution of a nontrivial small stratum step, let D_i be the real exceptional divisor defined by

The strict adjoint comparison makes D_i an effective nonzero divisor exceptional over both strata. Equality of the scaled pullbacks at its wall gives the exact Bott–Chern equality

For t in a later working interval all preceding factors on the right are nonnegative. Telescoping on a common resolution expresses the pullback of \{J\}+tA as the pullback of its nef working trace plus an effective divisor exceptional over that working stratum. Lemma 2.3 identifies the latter divisor with the negative part.

Push the resulting positive currents to one fixed smooth resolution of S and let t\downarrow0. Closedness of the pseudo-effective cone proves that J is pseudo-effective. The negative multiplicities converge valuation by valuation by the limit statement in Lemma 3.7. The comparison divisors have no strict prime of S, since the maps are small. Their coefficients also vanish at every place whose center meets the floor: each finite composition is unchanged on a neighborhood of the floor by the induction on smaller centers. Passing to the limit proves Equation (3.5).

The stratum has dimension less than n, and the divisor D_S constructed above is rational Cartier with support equal to its floor. Lemmas 3.11 and 3.12 give one small model V on which all transformed classes of \{J\}+tA are nef for 0\leq t\leq\delta. On a common smooth resolution p:W\to S,\ q:W\to V, define the real divisor F(t) in the common exceptional span by

Uniqueness of the representing exceptional divisor makes F(t) coefficientwise affine in t. It is effective on [0,\delta]: -F(t) is p-nef because the class pulled back from V is nef, so exceptional negativity applies. It is exceptional over V as well. Lemma 2.3 gives

Every divisorial negative multiplicity is thus affine on this whole interval. This remains true on higher resolutions by Lemma 2.4.

Choose a nontrivial stratum wall t_i\in(0,\delta). On a common resolution of S,V and the adjacent strata, their nef working models compute the same negative part on their respective open intervals. Their two affine expressions agree at t_i. But Equation (3.11) says that, at a prime with coefficient d_i>0 in D_i, their difference is

Their slopes differ by -d_i/t_i. A single affine function on [0,\delta] cannot agree with both on two open intervals. This contradiction excludes every such wall. Infinitely many walls would tend to zero, so the nontrivial diagrams on this stratum are finite.

There are finitely many surviving strata. Induction on their dimensions makes all sufficiently late ambient flipping and flipped loci disjoint from the floor. This proves the theorem.□

Proof. The signed representation persists under codimension-one pushforward. A curve disjoint from the transformed floor has degree zero for each component in this representation, and hence for the transformed adjoint. Every negative operation must therefore meet the floor. After the finite prefix supplied by Theorem 3.13 no further negative operation is possible. The continuation in Lemma 3.6 makes the last adjoint nef.□

Proof. These are, respectively, the two cases of [17, Lemma 2.6(1),(2)]. The first case is birational descent between spaces with rational singularities and does not require a projectivity criterion. The second case uses relative vanishing for the ordinary rational klt pair and the projective relative curve space. In particular, it can be applied to a projective log-Fano contraction before constructing generalized data on its base.□

Proof. Fix a Kähler class \omega on T and normalize \overline{\mathrm{NA}}(T) by \omega\cdot z=1. The resulting slice S is compact. Its unique point on R has negative D-degree, so d=c_1(D) is negative on a neighborhood of that point in S. On the complementary compact set, \alpha has a strictly positive minimum, while d is bounded. For all sufficiently small s>0, the class \alpha-sd is therefore strictly positive on S and is Kähler by cone duality.

Choose an integer m>0 making mD integral. Then

is Kähler. On a neighborhood in the base with a potential for \gamma, absorb that potential in a smooth metric on the same global line bundle -mD. Its curvature is positive on the fibres. The analytic positive-line-bundle criterion and the fibrewise relative-ampleness criterion make -mD relatively ample. The line bundle was global throughout, so no gluing of unrelated local polarizations is required.□

Proof. Suppose first that \pi is bimeromorphic. Grauert’s proper direct-image theorem makes \mathcal F=\pi_*\mathcal L coherent. On the nonempty open subset U^\circ where \pi is an isomorphism, it is a line bundle. Choose u_*\in U^\circ. Cartan’s theorem A says that global sections generate \mathcal F_{u_*}, so one global section has nonzero image in \mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}. Under

it gives a section s nonzero at the unique point above u_*. Its zero divisor is effective Cartier and represents \mathcal L. The section may vanish elsewhere; only its meromorphic inverse is being used to obtain a meromorphic trivialization.

For the general case, fix u\in U and y\in\pi^{-1}(u), and choose a \pi-ample line bundle \mathcal H. Relative Serre generation, after shrinking to a Stein neighbourhood of u, gives an integer n for which both \mathcal L\otimes\mathcal H^n and \mathcal H^n are relatively generated. Their proper direct images are coherent. Cartan’s theorem A and the evaluation maps supply sections s and t of these two bundles which are both nonzero at y. The quotient s/t is the required nonzero meromorphic section of \mathcal L. At no point is Y assumed Stein.□

Proof. Since f has a global relatively ample bundle and all its vertical curves lie on R, numerical invariance of ampleness on the projective fibres shows that -L is f-ample.

A local rational ordinary adjoint. Fix z\in Z and a relatively compact Stein neighbourhood U. Take a projective log resolution p:V\to T_U carrying \mathbf M, and put \rho=f p. Write its actual structure boundary as B_V, so

Every coefficient of B_V is less than one. Apply Lemma 3.17 to obtain a Cartier representative D_L of L on T_U. The same birational direct-image argument applies to the coherent rank-one reflexive canonical sheaf: it supplies a canonical form nonzero at a chosen point of the common smooth isomorphism locus. Use this meromorphic form on T_U and its pullback to V to choose compatible canonical representatives, and set

For every \rho-vertical curve, its degree equals that of \mathbf M_V. Thus N is \rho-nef.

The relative bigness of N uses the additional fact that \rho=f\circ p is bimeromorphic: both the small contraction f and the resolution p are bimeromorphic. It does not follow from relative nefness alone. In fact every real Cartier divisor is relatively big for this projective bimeromorphic map over the Stein neighbourhood. Here is the required ample-plus-effective decomposition explicitly. Write N=\sum_j r_jN_j, where r_j>0, \sum_jr_j=1, and the N_j are rational Cartier divisors in its finite Cartier span. Choose a \rho-ample line bundle and a Cartier representative P for it using Lemma 3.17. If q_jN_j is integral, apply the birational case of that lemma to \mathcal O_V(q_jN_j-P). Its nonzero holomorphic section has an effective Cartier zero divisor E_j, giving the actual relation q_jN_j\sim P+E_j. This use depends on \rho being bimeromorphic; the section is allowed to vanish along exceptional fibres. Therefore

Here H is \rho-ample and E\geq0, proving \rho-bigness. For a map with positive-dimensional general fibres the direct image used above can be zero; no such bigness claim is made for that setting. Make the finitely many section choices on a slightly larger Stein neighbourhood, and now shrink U so its closure is compact in that neighbourhood. Properness makes the inverse image of this closure compact. A log resolution of the finitely many resulting divisors therefore gives a uniform positive bound for the following choice. Choose \epsilon>0 small enough that (V,B_V+\epsilon E) is sub-klt. The divisor (1-\epsilon)N+\epsilon H is relatively ample. Express it as a positive combination of rational ample divisors and take general members of sufficiently high multiples. Relative Bertini [18, Theorem 2.20], applied on a log resolution of B_V+E, gives an effective \Theta\sim_{\R}N such that (V,B_V+\Theta) is sub-klt. The high multiples make the added coefficients arbitrarily small. All choices are made near the compact fibre; shrink U once to retain them.

Put \Delta=p_*(B_V+\Theta)\geq0. Write the actual linear equivalence as \Theta=N+\sum_j a_j\operatorname{div}_V(g_j). The bimeromorphic map p identifies meromorphic function fields, so g_j=p^*h_j. Pushing down gives K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j); in particular this divisor is real Cartier. Pulling back this same identity gives

Thus (T_U,\Delta) is klt. Any finitely many base line bundles in a relative linear equivalence can first be trivialized by shrinking U. Record the resulting equality as

using the positive support of \Delta. After a further relatively compact shrinking, the displayed divisors have finitely many prime components: their locally finite supports meet a compact inverse image. Equality of coefficients in (3.13) is a finite rational affine system in (d_i,t,b_j). The klt condition is open within this system, as seen on one log resolution; the pullbacks of its adjoints vary linearly by the displayed identity. A nearby rational solution therefore gives a rational effective klt boundary \Delta_q and t_q\in\Q_{>0} with

The individual local primes D_i need not be \Q-Cartier: the identity ensures that the total adjoint is \Q-Cartier.

One global flip algebra. The adjoint in (3.14) is f_U-antiample. Its ordinary flip and local finite generation follow from [31, Theorems 1.14 and 1.18]. Clearing the displayed linear equivalence identifies a Veronese of its canonical algebra with a Veronese of \mathcal R:=\bigoplus_{m\geq0}f_*L^m over U. Hence \mathcal R is locally finitely generated by [31, Lemma 2.26]. The graded pieces are coherent. Over a connected normal base open the inverse image is irreducible; products of nonzero sections of line bundles are nonzero there. Thus its graded section algebra is an integral domain, without choosing any global meromorphic frame of L. Compactness supplies one sufficiently divisible Veronese \mathcal R^{(r)} generated in degree one. Its relative Proj T^+ restricts to the ordinary flip over every such U. It is therefore normal, locally klt, and small over Z, and has the global relatively ample bundle \mathcal O_{T^+}(1). On the common big open this bundle is L^r. Reflexivity gives

This proves that L^+ is a rational line bundle before any assertion of factoriality. Relative positivity over the compact Kähler base makes T^+ compact Kähler.

For any global rank-one reflexive sheaf \mathcal F^+ on T^+, take its coherent reflexive transform \mathcal F on T using a common resolution. Some M=\mathcal F^{[m]} is a line bundle. Choose c\in\Q so that (M+cL)\cdot C=0, and clear its denominator to obtain an integral bundle J=n(M+cL) numerically trivial over f. The local rational klt pairs (3.14) have antiample adjoint, so Lemma 3.3 yields one global line bundle J_Z with J=f^*J_Z. On T^+, pull back J_Z, undo the rational twist ncL^+, and compare on the common big open. After clearing the already established index of L^+, reflexivity gives an invertible positive reflexive power of \mathcal F^+. This proves global strong factoriality.

The original generalized pair. The local log-Fano pairs above give R^if_*\mathcal O_T=0 for i>0 by [31, Theorem 5.2]. They make T locally klt, hence rational. Leray for a projective resolution then shows that Z is rational. Lemma 3.15(i) therefore gives f^*H^{1,1}_{\BC}(Z,\R)=R^\perp, with injective pullback. Thus A=t c_1(L)+f^*\eta for a unique \eta. Define A^+=t c_1(L^+)+(f^+)^*\eta; it is relatively Kähler.

To construct its actual discrepancy divisor, let \mathcal I be the ideal defined by

Principalize \mathcal I on a common resolution p:W\to T, q:W\to T^+ carrying \mathbf M. If \mathcal I\mathcal O_W=\mathcal O_W(-F_L), the tautological quotient gives the actual bundle identity

Hence E_L=F_L/r is effective and exceptional over both sides, and [E_L]=p^*c_1(L)-q^*c_1(L^+). If B_W is the original structure boundary, put B_W^+=B_W-tE_L. Then

Its pushforward is B^+, and its discrepancies do not decrease. The local ordinary comparison in (3.14) has discrepancy divisor t_qE_L, so its strictness proves the stated strictness for tE_L as well. Thus the original pair remains gklt.

Finally, write any \theta\in H^{1,1}_{\BC}(T,\R) uniquely as \theta=f^*\eta_\theta+s c_1(L) and set \theta^+=(f^+)^*\eta_\theta+s c_1(L^+). Its correction is the actual divisor sE_L. For a rational line M, the number s is rational. Applying Lemma 3.3 to a Cartier multiple of M-sL gives M=sL+f^*M_Z as actual rational lines. Reflexive extension on the common big open gives M^+=sL^++(f^+)^*M_Z. Consequently the cohomological transport agrees with the reflexive transform on Chern classes.□

Proof. Put D=K_T+B and fix a D-negative extremal ray R. On the finite positive support of B, effectiveness, the klt condition and negativity on R persist under small coefficient changes. Klt openness is checked on one log resolution of this support. Choose nearby effective rational klt boundaries B_0,\ldots,B_m and positive real numbers u_j with

For B=0 take just B_0=0. Every D_j is a global rational line bundle. The rational-step theorem for (T,B_0) provides a projective contraction f:T\to Z onto a compact Kähler base, contracting exactly R, with -D_0 relatively ample. The base is rational by [18, Lemma 8.8], so Lemma 3.15 gives f^*H^{1,1}_{\BC}(Z,\R)=R^\perp; use case (ii) when f is of fiber type. Let C be a rational curve spanning R, and set a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}. On every projective fibre, -D_j is numerically equivalent to the positive multiple -a_jD_0. Numerical invariance of ampleness on that fibre, followed by the relative-ampleness criterion, makes -D_j relatively ample. The positive combination -D is relatively ample as a real line bundle. This also handles a fibre-type contraction.

Suppose henceforth that f is birational. A Cartier multiple M_j=n_j(D_j-a_jD_0) is numerically trivial over Z and M_j-D_0 is relatively ample. Lemma 3.3, applied to the rational klt pair (T,B_0) gives a rational line bundle A_j on Z with the actual identity

Here and below an identity of rational line bundles means an isomorphism after a common integral multiple. The descent lemma is used only for birational f.

For a small f, let f^+:T^+\to Z be its rational D_0-flip. The rational theorem supplies a global relatively ample adjoint D_0^+ and preserves the stated factoriality and canonical-sheaf conditions. The other transformed adjoints D_j^+ are therefore rational line bundles. Transform (3.15) on the common big open and extend by reflexivity to obtain

All D_j^+ are relatively ample, as is D^+=\sum_j u_jD_j^+. Thus this same small map is the required ordinary real-boundary flip. The real ordinary discrepancy comparison proves that (T^+,B^+) is klt and gives the strict increases at exceptional centres.

For a divisorial contraction, write D_0=f^*D_0'+e_0E with e_0>0. Transforming (3.15) on the target and summing yields

This coefficient is positive. The same discrepancy comparison proves the required assertions. The ambient category was already preserved by the rational step in both cases.□

Proof. Let V be an irreducible positive-dimensional subspace of a reduced fibre. It is Moishezon. Normalize V and take a smooth projective modification \pi:\widetilde V\to V^{\mathrm n}, chosen so that an effective exceptional divisor E has -E relatively ample. The pullback of the Kähler class from V to its normalization is Kähler. Consequently \pi^*\omega-\delta[E] is Kähler for sufficiently small \delta>0. On the projective manifold \widetilde V the actual real line bundle \pi^*L-\delta E has the same curve degrees as that Kähler class, and is therefore ample. For example, subtract a sufficiently small positive multiple of a fixed ample class from the Kähler class; the corresponding real line bundle is nef by the projective numerical criterion. Also \pi^*L is nef, and the decomposition

shows that it is big. Thus

The real Nakai–Moishezon criterion for proper algebraic spaces [35, Theorem 1.6], applied to the algebraizations of the Moishezon fibres, proves that L is ample on every reduced fibre. The criterion is insensitive to nilpotents and reducible components. In the coefficient space of the finitely many global line bundles occurring in L, choose, for each fibre, a rational simplex containing L in its interior whose vertices restrict to ample rational lines on that fibre. Openness of fibrewise ampleness for each of these finitely many rational lines gives a base neighbourhood on which the entire simplex is relatively ample. Choose finitely many such neighborhoods covering the base and intersect their coefficient neighborhoods of L. Every rational point of the resulting neighborhood is relatively ample globally. A rational simplex in this intersection also expresses L as a positive combination of relatively ample rational lines. Clearing the denominator of one rational point proves projectivity.□

Proof. We use the smooth argument of [15, Theorem 3.1, Step 1]. Let d=\dim W-\dim S. Choose a rational class u\in H^2(W,\Q) close to a Kähler class. Write its Hodge decomposition as u=b+g^*v, where b is Kähler and v has types (2,0) and (0,2). Hodge type and the projection formula give

Since pushforward is rational, the class

is rational and of type (1,1). Lefschetz’s theorem makes it the Chern class of a rational line. The displayed equality supplies a metric with positive curvature locally over S, by adding a potential for the base class. Thus this line is relatively ample. For rationally connected general fibers, holomorphic two-forms are pulled back from the smooth base, so the hypothesis holds.□

Proof. The composition g is projective, since all the spaces are compact. Choose a g-ample line bundle on W, with integral Chern class H. Set d=\dim W-\dim S and

Thus c is the positive degree of H^d on a general fibre. The cohomological pushforward here is the usual pushforward between the smooth compact manifolds W and S, defined over \mathbb Q and of Hodge bidegree (-d,-d).

Define a rational linear operator on degree-two cohomology by

Every class of type (2,0) on W is pulled back from S, and the same holds for type (0,2). The projection formula therefore shows that \Pi kills both types. It preserves type (1,1). Consequently

By the Lefschetz (1,1) theorem the image consists of rational Chern classes of global line bundles. Applying \Pi to r^*\alpha gives

For example, express r^*\alpha in a rational basis of H^2(W,\mathbb Q) and apply \Pi to each basis vector. This gives the required finite real combination L_W.

Write L_W=\sum_j a_jL_j with actual line bundles L_j on W. For each j, let

It is a rank-one reflexive coherent sheaf. Strong \mathbb Q-factoriality gives an integer m_j>0 for which M_j=\mathcal F_j^{[m_j]} is a line bundle on T. The canonical identification over the isomorphism locus of r gives

for an integral r-exceptional divisor E_j. One may obtain this comparison directly from the evaluation map for r_*L_j: the two coherent rank-one sheaves agree away from the exceptional locus, so their invertible transforms differ by an exceptional divisor. No global meromorphic frame of L_j or of the canonical sheaf is required.

Set L=\sum_j(a_j/m_j)M_j and E=\sum_j(a_j/m_j)E_j. The preceding identities yield

The divisor E is r-exceptional and is numerically trivial on every r-contracted curve. The exceptional negativity lemma applied in both signs gives E=0. Injectivity of pullback by a resolution, or pushforward of the equality of currents, now gives \alpha=c_1(L)+f^*\gamma.□

Proof. The assumed decomposition makes analytic equivalence and line-degree equivalence identical on combinations of vertical curves. Their map into the analytic numerical space is therefore well-defined and injective. We must show that its closed effective cone is the entire face, including its classes represented by positive currents.

Fix z\in\mathcal F_f. For every \gamma\in H^{1,1}_{\BC}(S,\R), both a\omega_S+\gamma and a\omega_S-\gamma are Kähler for large a. Positivity then shows that z annihilates f^*\gamma. Fix a global relatively ample line H. If a real line D has nonnegative degree on every vertical curve, the projective relative numerical criterion makes D+\epsilon H relatively ample for every \epsilon>0. Its Chern class becomes Kähler after adding a sufficiently large multiple of f^*\omega_S. Therefore

Applying this to both signs of a relatively numerically trivial line shows that pairing with z factors through the finite-dimensional space of relative line degrees. It is nonnegative on its nef cone. By duality for this projective relative space, it is represented by an element of \overline{\mathrm{NE}}^{\mathrm{line}}(T/S). The decomposition in the statement makes its analytic image equal to z. Conversely every vertical effective curve lies in \mathcal F_f, and finite-dimensional injectivity preserves closedness. This proves the cone equality and the extremality claim.□

Proof. We reduce to fixed ordinary pairs, and construct one global program. Choose finitely many Stein open sets U_\ell\subset S and semianalytic Stein compact subsets W_\ell\Subset U_\ell whose interiors cover S. Such compacts are obtained by intersections with closed polydiscs in local embeddings of S. In particular they satisfy condition (P) of [31]: the source is normal, the base is Stein, the compact is Stein, and its intersection with any analytic subset defined near it has finitely many connected components. This remains true on every normal projective model. Shrinking U_\ell around W_\ell preserves the finite cover.

Actual ordinary replacements. On each chart represent the finitely many line bundles in the data by Cartier divisors. For a projective map to a Stein space this can be done by twisting a line bundle and its prospective meromorphic frame by sufficiently positive relative lines and using nonzero sections. The generic-rank-one direct-image argument is an alternative only when the map is bimeromorphic. Relative bigness gives N\sim_{\R}P+E, with P relatively ample and E\geq0. For sufficiently small \epsilon>0, (W,B_W+\epsilon E) is sub-klt near the compact inverse image. The line (1-\epsilon)N+\epsilon P is relatively ample. General divided relative sections give an effective representative T such that

All supports and coefficient margins are fixed near that compact. With compatible canonical representatives, push the finite principal-divisor expression for \Theta-N down and pull it back again. For \Delta_\ell=p_*(B_W+\Theta) this gives

In the last equality the chosen principal correction is included in the representative of K_X+\Delta_\ell. Thus (X_{U_\ell},\Delta_\ell) is ordinary klt and its boundary is relatively big. Lemma 11.15 of [31], with its big part equal to \Delta_\ell and its remaining part zero, replaces it, in its actual real linear equivalence class, by a klt boundary with an effective general relatively ample rational summand. Its condition on non-klt centres is vacuous. Choose the reserved ample summand to be \Q-linearly equivalent to a small rational multiple of a fixed global f-ample line. Its later strict transform is consequently \Q-Cartier by the transport of that global line, without a local factoriality claim.

We make these choices for a finite rational polytope of actual global line data. Start with a finite rational span containing J, and add relatively ample rational lines spanning the global relative degree space. The above construction is valid in a neighbourhood of J: retain a small part of the ample summand and use openness of ampleness and the strict klt inequalities. It is also valid along a segment from J to a sufficiently positive adjoint, by adding divided general ample divisors. Use finitely many such representatives and one log resolution on each chart. More explicitly, represent a finite simplex inside the open ample neighbourhood used above; varying its positive coefficients represents all nearby global line parameters on the same finite support. Form the joint finite affine system consisting of the divisor identities on all charts, with common global line coordinates and separate boundary and principal-divisor coordinates. This system is rational. Allow all boundary coefficients to vary, including those originally contributed by the real boundary B; fix only the reserved rational ample part and the zero-support constraints. Its projection to the global line coordinates contains the just constructed neighbourhood, so it has a rational affine section. Choose that section sufficiently close to the constructed real affine lift, by rational approximation in its affine space of sections, so that the strict klt and effectiveness margins are retained along the compact segment. In the joint solution space the strict klt and effectiveness margins permit a rational polytope containing the scaling segment. This supplies on every chart a rational polytope of ordinary klt boundaries with a fixed effective relatively ample rational summand, representing the restrictions of the same global line parameters. These boundaries, including the scaling representatives, are fixed before the program starts. Their divisors need not agree on different charts.

One global scaling construction. Apply the global degree-space construction in the proof of Proposition 3.4 to J and a general ample scaling direction in the chosen span. Here are the modifications that allow real line data and a nonbirational map to S. The global relatively ample line still gives a compact normalized slice of the curve cone. The fixed ordinary chart pairs give finite negative-ray truncations. A general direction avoids equal wall parameters for independent ray-degree functionals, exactly as in that proof. A positive wall therefore selects a single ray in the global degree space. A rational nef support is represented by a global rational line; a large multiple minus the chartwise ordinary adjoint is relatively ample. Ordinary relative base point freeness contracts the ray using the evaluation of this one global line.

If the contraction is small, choose a global rational line having negative ray degree and construct its flip by its one global graded algebra. The local ordinary proof of Proposition 3.18 applies to this algebra: all curves of the contraction have proportional degrees for global lines, so the detector and the driving line are positively proportional over the contraction; ordinary rational replacement gives local finite generation. This part of that proof needs only the already projective contraction and these line degrees. It does not require a prior assertion about the full Bott–Chern cone. The relative Proj is the next global model. In a divisorial step the exceptional prime and Lemma 3.3 give the same continuation as in Proposition 3.4. Killing the degree of any global line by a rational multiple of the detector, then applying that lemma, preserves global strong \Q-factoriality in both cases. The global lines continue to span the relative degree spaces. The rational support-simplex argument in that proposition makes the wall a descended relatively ample real line on its contraction base and gives a nonempty interval of ampleness immediately after each step. This argument uses relative line data and projectivity, not birationality of the map to S.

We check explicitly the persistence of the ordinary pairs. Expand the line parameters in finitely many global rational lines. Their reflexive transforms are actual rational lines by the preceding strong property. Pushing forward the fixed principal-divisor identities gives

Hence these ordinary adjoints are \R-Cartier; no local factoriality of individual prime divisors is being assumed. On a common resolution their comparison minus the appropriate positive multiple of the detector comparison is exceptional and relatively numerically trivial. Negativity in both signs makes this difference zero. Every chosen step is consequently negative for the fixed ordinary pairs. They remain effective and klt, and their boundaries remain relatively big. The same equality of comparison divisors, after adding the fixed base pullback, preserves the generalized pair and its discrepancy inequalities.

For termination, every already constructed working model is a weak log canonical model on each chart of a boundary in the fixed polytope: choose a parameter in its nonempty ample scaling interval. Indeed, actual descent at a preceding wall \lambda_j makes the comparison for J+tH equal to (1-t/\lambda_j) times its J-comparison. For a parameter in the current interval all preceding \lambda_j are at least t, so these comparisons are effective. Theorem E of [31] gives finitely many marked models on each chart. That theorem does not require locally \Q-factorial targets. There are therefore finitely many possible global marked working models. Indeed, isomorphisms between the restrictions of two existing marked models agree on their common dense open, and hence on overlaps. A repeated marked working model would have the same boundary transform and discrepancies, contrary to strict discrepancy increase at an intervening nontrivial step. The global program is finite. Relative pseudoeffectivity excludes a Mori fibre endpoint, so J_m is nef over S. This is a finite-cover argument for a single constructed program; it does not glue independent minimal models.

Semiampleness and the global polarization. On the endpoint the fixed ordinary pair is klt, its effective boundary is relatively big, and its adjoint is nef over W_\ell. Lemma 11.16 of [31] applies to this already existing weak model. More explicitly, Lemma 11.15 replaces the endpoint boundary by A_\ell+B_\ell, where A_\ell\geq0 is relatively ample, B_\ell\geq0, and the pair is klt. Theorem 8.3 then gives semiampleness near W_\ell: its log canonical, klt-existence, real Cartier, ample-summand and nef-over-compact hypotheses have all been checked.

The actual equivalences identify these local semiample adjoints with restrictions of J_m. Their ample models agree on overlaps by [31, Lemma 11.3]. To retain a global polarization, shrink the initial parameter polytope around J after the finite program: all its finitely many strict step signs persist, so the transformed ordinary boundaries remain klt and big. On each chart the pushforward of its initial common ample rational summand is effective and relatively big. Apply [31, Lemma 11.13] to this common big rational part and the small endpoint boundary polytope. All pairs are klt, so the condition on non-klt centres is vacuous. The lemma gives a common general relatively ample rational summand by one \Q-linearly trivial translation. In particular the rational parametrization is preserved; separate applications of Lemma 11.15 to individual parameters would not suffice here. In this finite rational span Theorem 11.17 of [31] makes the endpoint nef region a rational polytope. Intersect the finitely many inverse images of these regions and express J_m as a positive combination of rational points in its minimal face. The corresponding global rational lines are semiample on every chart by the same ordinary base point free argument. After clearing denominators and taking common sufficiently divisible powers over the finite cover, each is relatively generated globally: generation on the charts is generation of its global coherent relative evaluation map. The product of these finitely many relative morphisms, followed by Stein factorization, gives a projective g:X_m\to Z, a projective \rho:Z\to S, and descended global lines whose positive combination H is relatively ample. It realizes the glued ample model and gives J_m\sim_{\R}g^*H. Adding back f_m^*\beta proves the asserted Bott–Chern identity. Composing the effective step comparisons proves the final discrepancy statement.□

Proof. Lemma 11.15 and Theorem 8.3 of [31] give local semiampleness. For an existing weak model this is also Lemma 11.16. To globalize the polarization, take a finite rational span of the global lines defining J. On each chart the ample part of the normalized boundary allows small perturbations in this span. The joint rational affine construction in the preceding proof gives a rational family with a common rational ample part; all boundary coefficients are allowed to vary. Theorem 11.17 then gives a rational nef region on each chart. The finite-intersection, rational-vertex and global-evaluation argument in that proof gives one projective contraction and its actual descended ample real line. Only its local ample models are identified by uniqueness.□

Proof. The line N is fixed; it need not be a general scaling direction. In particular several extremal rays may have the same wall parameter. Choose \epsilon>0 such that D+2\epsilon N is still not pseudoeffective over S. On the finite Stein-compact cover choose divided general effective representatives of N so that all the resulting ordinary pairs representing D+tN, 0\leq t\leq T, are klt. For \epsilon\leq t\leq T reserve a common effective relatively ample rational summand. The joint rational-family construction in Proposition 3.25 puts these fixed pairs in the polytopes required by Theorem E of [31].

Suppose a finite prefix has reached X_i, and put \mu_i=\lambda_{i-1}, with \mu_0=T. Its transformed D_i+\mu_iN_i is nef. Define

The feasible set is a nonempty closed interval. Inductively \lambda_i>2\epsilon: all previous steps have nonpositive comparison for D+2\epsilon N, so its transform is still not pseudoeffective. A first threshold at most 2\epsilon would make this transform nef by convexity between the current and preceding thresholds, a contradiction.

There is a D_i-negative extremal ray R_i with (D_i+\lambda_iN_i)\cdot R_i=0. To see attainment without a generality assumption, let t<\lambda_i tend to \lambda_i. A ray negative for D_i+tN_i has positive N_i-degree, because D_i+\mu_iN_i is nef. When t>\lambda_i/2, it is also negative for D_i+(\lambda_i/2)N_i. The latter has fixed ordinary klt big-boundary representatives on the charts, since \lambda_i/2>\epsilon. Normalize each of these boundaries by [31, Lemma 11.15] as A_\ell+B'_\ell, with A_\ell\geq0 relatively ample and the new pair klt. The A_\ell-truncation of the cone theorem for K+B'_\ell gives finitely many rays negative for the entire adjoint in question. The finite cover therefore gives finitely many global negative rays. One of these rays attains the wall, as required.

Choose one such ray, even if the wall vanishes on other rays. The rational-support argument for an individual negative ray in Proposition 3.4 gives a global rational nef support annihilating exactly R_i, and ordinary relative base point freeness contracts it. For the real ordinary boundary one may first choose a nearby rational klt boundary that is still negative on R_i. The contraction therefore has a rational ordinary antiample adjoint. Its flip, when small, is constructed from one global detector algebra as above.

If the contraction is of fibre type, stop. Otherwise the wall w_i=D_i+\lambda_iN_i has degree zero on R_i. In the finite rational span of its actual line factors, this zero-degree subspace is rational: its defining functional has rational values on rational lines. Thus w_i is a real combination of rational lines of zero ray degree. Lemma 3.3 descends those lines to the contraction base. Their combination is nef over S, by lifting curves through the projective contraction. Pulling it to the positive side shows that the transformed wall stays nef. The next threshold therefore satisfies \lambda_{i+1}\leq\lambda_i; equality is permitted. No interval of ampleness between successive thresholds is required.

If E_i\geq0 is the ordinary D_i-comparison on a common resolution, this actual wall descent gives, for every real t,

For t\leq\lambda_i the error is effective. This proves all the nonpositivity and fixed-pair persistence assertions used in the induction, including the lower bound on thresholds.

At every working model use the wall parameter t=\lambda_i, rather than an interior ample parameter. Since \lambda_i\leq\lambda_j for every earlier step, the displayed comparisons show that X_i is a weak log canonical model of the initial fixed chart pairs representing D+\lambda_iN. Its trace is nef by definition. Theorem E counts all these marked weak log canonical models, including the ones at repeated wall parameters. The finite chart cover gives a finite global marked list. Repetition would contradict the strict ordinary D-discrepancy increase at an intervening step. Thus this program with the specified N is finite. Non-pseudoeffectivity excludes a nef endpoint for D, so the last contraction is a Mori fibre contraction. The comparison at t=1=\lambda_i is crepant: such a step supplies no strict discrepancy improvement for D+N, while it is still strictly negative for D.□

Proof. Choose a common projective log resolution p_0:U_0\to X of the original nef carrier and the prescribed modification, and write

The class \mathbf M_{U_0} is nef, and the negative part (C^0)^- is exceptional over X. The modified-big preparation [39, Lemmas 2.10 and 2.23] can be made with the prescribed resolution. We give its divisor and ideal details. Choose an effective divisor E_0 with sufficiently large coefficients and full p_0-exceptional support so that the class

is big. Resolve a Kodaira decomposition of this class. On a further projective modification a:U\to U_0 with U smooth one obtains

where \kappa is Kähler. The support can be made SNC together with all the other divisors. On any additional blowup, subtract a sufficiently small effective exceptional class from the pulled-back Kähler class and add it to the divisor; an exceptional divisor whose negative is relatively ample makes the resulting class Kähler again.

Put p=p_0a and

Then

and every negative component of C^{\mathrm{alt}} is p-exceptional. Let C^{\mathrm{old}} be the crepant structure boundary of the original data on this same model. For 0<t\ll1, set

The class \beta is Kähler, the adjoint identity is unchanged, and negative coefficients still occur only on exceptional divisors. There are only finitely many coefficients. Thus t can be chosen so that

Pushing forward the corresponding inclusion of invertible sheaves gives (3.17). The Kähler class \beta plus the strict transform of the effective trace boundary is big; its pushforward is the prepared total boundary. If the original pair is gklt, its multiplier ideal is the unit ideal, and hence so is \mathcal J_1.□

Proof. Choose a projective resolution \mu:W\to Z, with W smooth compact Kähler and reduced exceptional divisor F having SNC support. Apply Lemma 3.29 on a carrier dominating the main component of the base change. We obtain projective maps p:U\to X and q:U\to W with fp=\mu q and

Here U is smooth compact Kähler, \beta is Kähler, and all negative components of C are p-exceptional. The maps are projective by composition and factorization over the separated base W. The map q has connected fibres: this holds over the open set where \mu is an isomorphism, and its Stein factorization is finite and bimeromorphic over the normal space W.

The integral divisor -\lfloor C\rfloor has class

The pair (U,\{C\}) is klt with SNC boundary. Relative Kawamata–Viehweg vanishing, first for p and then for fp, applies because the positive class modulo pullbacks is Kähler [39, Theorem 2.2]. Equivalently, the real line bundle -K_U-C is relatively ample, so the vanishing follows from [32, Theorem 1.2] with the identity as its first morphism. Leray gives R^if_*\mathcal J_1=0 for i>0. Applying f_* to the sequence defining N_1, and using f_*\mathcal O_X=\mathcal O_Z, proves (3.18). If N_1 dominates Z, the ideal f_*\mathcal J_1\subseteq\mathcal O_Z vanishes generically and therefore vanishes. This proves that alternative.

Suppose from now on that N_1 does not dominate Z. Every q-horizontal coefficient of C is then less than one. Indeed, a component with coefficient at least one forces the multiplier ideal to be proper along its image in X, and its image would make N_1 dominate Z. If Z is a point, the asserted base presentation is the dimension-zero presentation. We therefore assume b=\dim Z>0.

The fibre metric. Choose a smooth representative \rho of \mu^*\gamma-c_1(K_W), a Kähler form \omega_W, and a Kähler form representing \beta, still denoted by \beta. For some a>0,

Consider the holomorphic line bundle

Equation (3.21) gives a singular Hermitian metric h on L whose curvature current is

Its weights are smooth functions plus the divisorial logarithms with coefficients in \{C\}. We normalize dd^c so that dd^c\log|z|^2=[z=0] and write squared-norm weights as e^{-\varphi}. Let e be the canonical meromorphic section of

so that \operatorname{div}(e)=-\lfloor C\rfloor.

Choose an analytic Zariski open set W^\circ\subseteq W on which q is a submersion, the horizontal boundary is relatively SNC, there are no vertical boundary components, \mu is an isomorphism onto an open subset of Z, and coherent base change holds in degree zero. On q^{-1}(W^\circ), the divisor -\lfloor C\rfloor is effective and exceptional over X. Normality and the contraction property therefore identify

Here K_{U/W} is viewed as the relative canonical line bundle. The fibre norm

is positive and finite. The fractional SNC weights are integrable; any zeros of e_w only improve integrability. Relative SNC coordinates and a partition of unity show that I is smooth: parameter derivatives act on smooth factors multiplying fixed integrable monomial weights.

Set \psi=-\log I on W^\circ. Then

Here are the details for the singular weights. Locally choose \ell with dd^c\ell=\rho-a\omega_W, and put h'=he^{-q^*\ell}. Its weights are psh, and its fibre norm satisfies I'=e^{-\ell}I. Restrict the family to a small coordinate line disk \Delta_R, with centre 0. The sharp L^2 extension theorem gives a twisted top form extending e_0 and satisfying

Top-form norms here are normalized to use Euclidean area in the base coordinate. We use the Stein hypersurface form of the optimal extension theorem [37, Theorem 2.2 and §3.5, (3.1)], with defining function the disk coordinate divided by R.

To check applicability, embed the restricted projective family into projective space over the disk and remove a relative hyperplane section that does not contain the central fibre. The complement is Stein. A holomorphic section of L on this Stein space can be chosen nonzero at a point of the central fibre; deleting its zero divisor trivializes L and leaves a Stein space. The central fibre is smooth and connected, hence irreducible, so both deletions leave a dense open subset of it. For a zero-dimensional fibre choose the deletions to avoid its point. In the resulting trivialization the weight is psh, and the sharp extension theorem applies to canonical forms. The extension extends across the deleted hypersurfaces in reverse order by local L^2 removability: in every original frame the metric is bounded below by a smooth positive metric. The extension retains its norm bound and its prescribed restriction to the central fibre.

After dividing E by the base differential, the rank-one direct image identifies it with b(t)e, where b is holomorphic and b(0)=1. Thus the disk average of |b(t)|^2I'(t) is at most I'(0). Jensen’s inequality and the submean inequality for \log|b|^2 give

The same argument applies to every sufficiently small disk at every centre and in every complex line direction. Hence -\log I'=\psi+\ell is psh, which proves (3.23).

Extension over the base. Let D be a prime divisor in W\setminus W^\circ. Choose a component Q of q^*D dominating D. When D is not \mu-exceptional, choose Q not p-exceptional: take a component over the corresponding prime on Z already on X and then its strict transform. Put

At a general point of Q, avoiding all other relevant divisors, there are local coordinates with

Indeed the map Q\to D is submersive at a general smooth point, and a local unit in the first equation is absorbed by a holomorphic uth root. Such points can be chosen above every point of a dense analytic open subset of D by generic smoothness and properness. Integration over a fixed small polydisk in the remaining fibre coordinates gives

for some c_*>0 near each such base point. The section and metric contribute the exponent c, while division by the base differential in the relative canonical form contributes u-1.

For a nonexceptional D the chosen Q has c\geq0, so (3.25) bounds \psi from above near its general points. For the finitely many exceptional divisors, the same estimate supplies a nonnegative integer m such that, if k is a smooth metric on \mathcal O_W(F) and s_F its divisor section,

is locally bounded above near general points of every divisor. Adding local potentials of \rho+m c_1(k)-a\omega_W to \varphi gives psh weights by (3.23). These weights extend across the divisors and then across the remaining analytic set of codimension at least two, giving

on all of W.

A Kähler carrier. Regularize the current in (3.26) in its class, retaining a lower bound for the potentials. More precisely, choose an approximation with analytic singularities such that

These simultaneous properties are the Bergman regularization properties of [21]; see also [60, pp. 59–60]. Resolve the analytic singularities and the support of F by a sequence of blowups with smooth centres r:T\to W, with simultaneous SNC support, and write

The residual current \zeta has locally bounded potentials. Removing the divisorial part from the positive current r^*S'-(a/2)r^*\omega_W shows that

The difference has vanishing Lelong numbers, so its class is nef by regularization. Choose an effective r-exceptional divisor G with -G relatively ample. For 0<\eta\ll1, the class (a/2)r^*\omega_W-\eta[G] is Kähler. Therefore

has Kähler \beta_T. If r is an isomorphism, omit G and its correction. The choice of \eta will be decreased once more below.

The exact class calculation is

All possibly negative terms in C_T are \tau-exceptional. Consequently B_Z=\tau_*C_T is effective. Specify the nef b-data by \mathbf N_T=\beta_T; the displayed identity defines generalized data on Z with adjoint class \gamma. At the level of currents, pushing the identity gives the required trace identity on Z: exceptional divisor currents push to zero, and a dd^c difference pushes to a dd^c difference. Canonical traces can be computed using local meromorphic canonical frames; a change of frame changes the corresponding Chern potentials by the usual logarithmic transition term. Thus only the whole adjoint, rather than its individual traces, needs to be a Bott–Chern class. The class \beta_T plus the strict transform of B_Z is big on T, so its pushforward B_Z+\mathbf N_Z is globally modified big. These are (3.19).

The multiplier ideal. Take a local holomorphic function v\in f_*\mathcal J_1. By the pushforward description of \mathcal J_1,

Thus q^*\mu^*v\,e, tensored with a local base canonical frame, is a holomorphic L-valued top form on U. Its squared norm is locally integrable by the fractional SNC weights in (3.22). Over the submersion locus, fibre integration followed by base integration is precisely this top-form integral. Properness and a finite coordinate covering therefore give

locally on W; the integral is taken where the original fibre metric is defined.

The lower bound on \varphi' in (3.27) preserves this integrability after replacing \psi by \varphi'-m\log|s_F|_k^2. Change variables through r. Along a prime divisor of T, let h,j,k_D,d be respectively the coefficients of H, \operatorname{div}(\tau^*v), K_{T/W}, and r^*F. The local integrand has exponent 2(j-h+md+k_D), up to a bounded positive factor. Its integrability implies j-h+md+k_D>-1, or, since j is integral,

Therefore

Choose \eta>0 sufficiently small that adding \eta G does not change any of these finitely many round-downs. The resulting inequality is exactly \operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor, proving (3.20) as an inclusion of ideal sheaves. In particular, if f_*\mathcal J_1 is the unit ideal on an open set, then so is \mathcal J_Z, and the base data are gklt there.

Finally, the nef data descend to T, and the structure boundaries on higher models are defined by crepant pullback of (3.19). On every common refinement of the source and base carriers, both adjoints are the pullback of \gamma. Thus the identities remain exact on higher models.□

Proof. Write h:D\to S for a Moishezon contraction defined by \delta. For a closed subspace D_0\subset D, take the Stein factorization

of its restriction. The first map is a contraction, and the second is finite. Thus a curve in D_0 is contracted by the first map exactly when its \delta-degree is zero. Choose a projective surjection P\to D witnessing that h is Moishezon, with P\to S projective. The closed subspace P_0=P\times_D D_0 is projective over S and projective and surjective over D_0. It is also projective over S_0: a relatively ample line for P_0\to S remains ample on the fibres of P_0\to S_0. This proves the first assertion, including its scheme structure.

For the descent direction of the second assertion, every curve in a fibre of g has \delta-degree zero. A connected projective complex space is connected by chains of compact curves after passing to its reduction. Therefore h is constant on the underlying set of each fibre of g. It factors as h=k\circ g for a holomorphic map k:V\to S. To see that this factorization holds also for nonreduced spaces, choose a coordinate neighbourhood in S of the image of a fibre. Properness permits restriction to a neighbourhood of its point in V whose inverse image maps into that coordinate neighbourhood. The coordinate functions of h then descend uniquely through g_*\OO_D=\OO_V; their analytic relations descend through the same equality. These local factorizations glue. Furthermore,

Every irreducible compact curve C\subset V is dominated by a compact curve C'\subset D. Indeed, after base change to the normalization of C, take a component dominating that curve and a projective multisection. The projection formula gives

Consequently k contracts exactly the curves of \epsilon-degree zero. The composite P\to D\to V is a projective surjection, and P\to S is projective, so it witnesses that k is Moishezon.

Conversely, suppose k:V\to S is a Moishezon contraction defined by \epsilon. Then k\circ g is a contraction, and the projection formula verifies its curve criterion, treating separately curves contracted by g. If Q\to V is a projective surjection witnessing that k is Moishezon, then D\times_V Q\to D is projective and surjective, while D\times_V Q\to Q\to S is projective. This supplies the required witness.□

Proof. Take a projective common log resolution r:U\to X carrying the nef datum, and put s=g\circ r. By crepancy there is one real SNC divisor C such that

where \beta is globally nef. The multiplier ideals are

Their direct-image equality follows at once. For vanishing, set L=\OO_U(-\lfloor C\rfloor) and \Delta=C-\lfloor C\rfloor. The pair (U,\Delta) is klt, and the actual real line

has class \beta-s^*A_Y. It is relatively nef over Y and relatively big because s is bimeromorphic. The same statements hold over X. Relative Kawamata–Viehweg vanishing, in the form of [32, Theorem 1.2] with the first morphism the identity and the real line \mathcal H=-K_U-C, therefore gives

The real-line identity in that theorem is the displayed definition of \mathcal H; its relative positivity is checked by the class identity above. The klt condition means that there are no proper log canonical strata to test for relative log bigness. Leray gives R^ig_*J_X=0 for i>0. Pushing forward 0\to J_X\to\OO_X\to\OO_{N_X}\to0 and using g_*\OO_X=\OO_Y now proves the asserted structure-sheaf equality. It also identifies the scheme-theoretic image with N_Y, giving the projective contraction between these closed subspaces. Apply Lemma 3.31 to conclude.□

Proof. The inclusion J_o\subset J_1 gives N_1\subset N_o as closed analytic subspaces. By Lemma 3.31, the given contraction restricts to N_1. Proposition 3.30 identifies

Thus the restriction N_1\to N_* is a projective contraction. The restricted adjoint is the pullback of \gamma|_{N_*}, so Lemma 3.31 descends the contraction to N_*. Nonnegative degrees on curves of N_* follow from projective multisections and nefness of A.

If N_1 dominates the normal irreducible space Z, then I=0: a local function in I pulls back to zero on the dominating closed subspace and hence vanishes generically on Z. Normality, and in particular reducedness, gives its vanishing. Therefore N_*=Z, which proves the assertion in this case.

If N_1 does not dominate Z, the constructed base ideal satisfies I\subset J_Z. Hence

as closed analytic subspaces. Restriction by Lemma 3.31 provides the required contraction on V(J_Z), with its complete scheme structure. The generalized pair on Z has globally nef carrier data and globally modified-big total boundary by Proposition 3.30. When the stated nef and NQC conditions hold, these facts are precisely the hypotheses of \mathsf C_{\dim Z}, which gives the contraction on Z. Finally Lemma 3.31 pulls it back through the projective contraction f to X. The case of an empty non-klt closed subspace has the same interpretation, with its contraction condition vacuous.□

Proof. Properness, normality and connected fibres identify both f_*\mathcal O_X=\mathcal O_Z and f_*\mathcal O_X^*=\mathcal O_Z^*. Apply Rf_* to the analytic exponential sequence. The local surjectivity of \mathcal O_Z\to\mathcal O_Z^* and the stated coherent vanishing give

Let \xi\in H^2(X,\mathbb Z) have zero degrees on contracted curves. A germ of its edge image in R^2f_*\mathbb Z_X is, under this isomorphism, represented after shrinking U by an actual line bundle L on X_U. Equality of these germs means that, after further shrinking, c_1(L) and \xi|_{X_U} have the same class. In particular L has degree zero on every curve in every fibre above this smaller neighbourhood.

On a sufficiently small Stein base neighbourhood, twisting by a relatively ample line and taking a quotient of two nonzero relative sections gives a meromorphic Cartier representative for L. Thus the Cartier-divisor form of base point freeness applies. The line bundle L is relatively nef, and aL-(K_{X_U}+\Delta_U) is relatively ample for every real a. The projective analytic base-point-free theorem applies to the Cartier line L. Its conclusion is generation for every sufficiently large integer power, not only for a divisible subsequence [31, Theorems 6.2 and 6.5]. Thus, over a further relatively compact neighbourhood, both L^m and L^{m+1} are relatively generated. A generated line of degree zero on every curve of a projective connected fibre defines a constant projective map on that fibre: the restriction of the generated line is the pullback of \mathcal O(1), and a positive-dimensional projective image contains a curve of positive degree. The corresponding relative image is therefore finite and bimeromorphic over the normal base, and is the base itself. Consequently

for line bundles on the neighbourhood in Z. Taking their quotient proves L\simeq f^*(M_{m+1}\otimes M_m^{-1}). Its germ in R^1f_*\mathcal O_X^* is zero, so the edge image of \xi is zero.

The integral Leray spectral sequence, using R^1f_*\mathbb Z_X=0, gives

Exactness at the middle term proves the assertion. The argument is integral throughout and does not discard torsion. The converse follows by restriction of a pulled-back class to a fibre.□

Proof. Positivity of the coefficients and nonnegativity of the summands imply that every \gamma_{ij} annihilates the contracted ray. Apply the integral descent lemma to the chosen integral lift of m_j\gamma_{ij} and pull its descended class back to X_{i+1}. This keeps m_j fixed.

To verify nonnegativity, take a curve on X_{i+1}. If it is contracted by f_i^+, every descended summand has degree zero. Otherwise its image is a curve \Gamma\subset Z_i. The projective map f_i has a compact curve mapping onto \Gamma with positive degree: take a component of its inverse image dominating \Gamma and intersect with sufficiently many relatively ample hyperplanes. Nonnegativity of the old summand on this curve implies nonnegativity on \Gamma, hence on the curve on X_{i+1}. This argument uses a positive-degree lift, not a degree-one section. Finally m_j\gamma_{ij}\cdot C is a nonnegative integer, giving the stated lower bound.□

Proof. A negative ray is K_{X_i}-negative because \alpha_i is nef. Choose its rational curve generator with the ordinary length bound. If its \alpha_i-degree were positive, then

a contradiction. The step is thus an ordinary canonical step with \alpha_i pulled back from its base, and the preceding lemma applies. On a common resolution the pullback of \alpha_{i+1}-c_1(K_{X_{i+1}}) is the pullback of \alpha_i-c_1(K_{X_i}) plus the effective canonical comparison. Thus it stays big. Equality of the crepant pullbacks of \alpha_i preserves its nonbigness. The identity

excludes pseudo-effectivity of its first summand. A nef endpoint is therefore impossible.□

Proof. For a modification of a smooth compact Kähler manifold the two splittings are the degree-two modification formula. Suppose that they hold for X=X_i. We use only the following birational descent fact: for a proper bimeromorphic map between normal compact spaces in Fujiki’s class with rational singularities, pullback is injective on H^2(-,\mathbb R) and on Bott–Chern cohomology, and its image in either group consists of the classes having degree zero on every contracted curve. This is the bimeromorphic case of [17, Lemma 2.6(1)].

First note that a degree-two class \xi\in H^2(X,\mathbb R) has the same degrees on curves as a Bott–Chern class. Indeed, pull it to a smooth projective resolution a:W\to X and take the (1,1)-part of its Hodge decomposition. The induction hypothesis writes that part as a^*\beta+[F], with F exceptional over X. On every a-contracted curve, a^*\xi and its (2,0) and (0,2) parts have degree zero. Hence F has degree zero on every such curve. Exceptional negativity applied to both signs gives F=0. Projective multisections lifting a curve of X then show that \xi and \beta have the same degrees on every curve of X.

Suppose first that X\dashrightarrow X^+ is a small detected flip. Write f:X\to Z and f^+:X^+\to Z, and fix an integral curve C on its negative ray. For \xi\in H^2(X,\mathbb R) set

By the preceding paragraph, \xi-s c_1(L) vanishes on every f-contracted curve. Thus it is f^*\eta for a unique \eta\in H^2(Z,\mathbb R). Define

For a Bott–Chern class, the same descent takes place in Bott–Chern cohomology, so the two transforms agree. The detected flip construction gives an effective rational divisor E_L on a common projective resolution a:W\to X, b:W\to X^+ such that

Consequently

If \xi is rational, then s is rational. Since the image of f^* on real degree-two cohomology is the realification of a rational subspace, its unique preimage \eta is rational as well. This proves rationality of T\xi.

The sets of prime divisors exceptional for a and b are identical: the two working spaces are small bimeromorphic. The old splitting and the displayed pullback identity therefore span

and similarly in bidegree (1,1). The sum is direct. If a pullback from X^+ equals an exceptional divisor class, that divisor has degree zero on all b-contracted curves, so both signs of negativity make it zero; pullback injectivity then makes the original class zero. In particular the dimensions on the two sides agree, T is injective by the old direct sum, and T is surjective. The rational version follows either from the same argument or from realification.

For a divisorial contraction f:X\to X^+, let D be its exceptional prime. Its degree on the contracted ray is negative: otherwise D would be f-nef, contrary to exceptional negativity. For any \xi, subtract

and descend the result by the same birational cohomology lemma. This defines T\xi. On a common resolution its pullback difference is a multiple of the total transform of D. The target’s exceptional prime set is the old one together with the strict transform of D. The same spanning and directness argument gives the target splitting and surjectivity; its kernel is \mathbb R[D]. Rationality is again immediate from the quotient of rational intersection numbers.

These arguments initially prove the splitting on a chosen common resolution. To obtain it on any projective smooth resolution U\to X^+, dominate that resolution and the chosen one by a smooth common projective resolution. Apply the smooth degree-two modification formula upstairs and push down to U. An exceptional divisor pushes either to zero or to a divisor exceptional over X^+. This proves spanning on U; negativity gives directness there.

Finally let \mathcal H be a holomorphic line on U and put \mathcal Q=(p_*\mathcal H)^{**}. Global strong \mathbb Q-factoriality gives an m>0 for which \mathcal M=\mathcal Q^{[m]} is invertible. The coherent evaluation comparison gives

for an integral p-exceptional divisor F. This comparison is obtained from the coherent direct image and evaluation map; it does not assume that \mathcal H has a global meromorphic section. Dividing first Chern classes by m proves the line assertion.□

Proof. Choose any projective smooth resolution q:W\to Y. The composite p=rq is a projective resolution of X. Its exceptional primes are exactly the q-exceptional primes together with the strict transforms \widetilde E_j of the E_j. Since every E_j is \mathbb Q-Cartier,

for a rational q-exceptional divisor F_j. Substitute these identities in the splitting for p. The result spans H^2(W,\mathbb R) by q^*H^2(Y,\mathbb R) and the q-exceptional divisor classes. Their sum is direct: an exceptional divisor whose class is a pullback has degree zero on every q-contracted curve, and applying exceptional negativity to both signs makes the divisor zero. Pullback injectivity then applies. The same argument in bidegree (1,1) and over \mathbb Q proves the splitting property for Y.

Now expand q^*\xi, for \xi\in H^2(Y,\mathbb R), using the original p-splitting and replace the [\widetilde E_j] as above. The difference between q^*\xi and a class in q^*(r^*H^2(X,\mathbb R)+\sum_j\mathbb R[E_j]) is q-exceptional, and is therefore zero by the direct sum just proved. Thus the displayed formula for H^2(Y,\mathbb R) spans. Its directness follows from exceptional negativity for r and pullback injectivity. The rational and Bott–Chern statements follow by the identical argument. The line bundle assertion of the preceding lemma uses only global strong \mathbb Q-factoriality and hence applies to Y as well.□

Proof. Rational classes modulo the fixed base. Choose an f_0-ample line bundle with integral Chern class H, and put e=\dim U-\dim Z and c=f_{0*}(H^e)>0. The rational operator

preserves type (1,1) and kills types (2,0) and (0,2). Indeed these latter classes are pulled back from Z, and the projection formula applies. Thus \Pi(H^2(U,\mathbb Q))\subset H^{1,1}(U)\cap H^2(U,\mathbb Q). By the Lefschetz (1,1) theorem, choose an actual line bundle L_{j,0} and an integer m_j>0 with

The original summands \eta_{j,0} need not have type (1,1). Their base components account for this. Their weighted sum gives

The final assertion follows because f_0^* is an injective Hodge map and the other terms have type (1,1). The identity is therefore also one of Bott–Chern classes. Each L_{j,0} has nonnegative integral degree on every Z-vertical curve.

Induction across an actual contraction. Suppose at a finite stage f_i:U_i\to Z we have actual line bundles L_{j,i}, nef over Z, and

Assume also that \alpha_i is nef and that the transformed D_i is generalized klt. These assertions hold initially. Let R be a (D_i+a\alpha_i)-negative extremal ray over Z. It is D_i-negative because \alpha_i is nef. Lemma 3.23 and Lemma 3.24 identify the relative curve cone with the corresponding face of the full analytic cone, at the initial model and after every step. The analytic cone theorem therefore supplies a rational generator C with

All L_{j,i}\cdot C are nonnegative integers. If \alpha_i\cdot C>0, one of them is at least one, and hence \alpha_i\cdot C\geq\delta. Therefore

a contradiction. We have \alpha_i\cdot C=0 and L_{j,i}\cdot C=0 for every j. Every curve contracted by the projective extremal contraction h_i:U_i\to Y_i has class on R when tested by global line bundles, so the same vanishing holds for all of them.

On each fixed Stein chart of Z, choose the effective real ordinary representative of the initial relatively ample nef datum once, before running the program. Its actual real linear equivalence to the fixed real line-bundle representative persists under pushforward. The Bott–Chern comparison with the generalized adjoint modulo the fixed base class persists separately by exceptional negativity, as detailed below. Thus every h_i is, locally on Y_i, a contraction with an ordinary real klt log-Fano boundary. On a relatively compact base chart, rational approximation in the finite affine system of Cartier adjoint identities gives a rational effective klt boundary with the same antiample sign, as in Proposition 3.18. Lemma 3.3 therefore applies. Consequently

For a divisorial contraction set L_{j,i+1}=M_{j,i}. For a flip h_i^+:U_{i+1}\to Y_i, set L_{j,i+1}=(h_i^+)^*M_{j,i}. These are actual line bundles with the same integers m_j. The descended Bott–Chern class

satisfies h_i^*\alpha_{Y_i}=\alpha_i. It is nef by descent of nefness through the projective surjection h_i; its pullback to the next model is \alpha_{i+1}. This proves that the step is crepant for \alpha_i. Hence it is also a D_i-negative step, and the original generalized pair remains gklt.

Finally each M_{j,i} is nef over Z. For a Z-vertical curve B\subset Y_i, projectivity supplies a curve B'\subset U_i mapping onto it with some positive degree d_B. Then

No assertion d_B=1 is required. Pullback preserves this relative nefness, so the induction applies on U_{i+1}. Its curve degrees are again integers because the transported objects are line bundles, not because any curves were lifted with degree one. This proves the fixed bound through the whole program.□

Proof. Take a common projective resolution p:W\to U, q:W\to V. The negativity comparison for the finite program is

This comparison is valid for a relative negative program: its proof uses the negativity of the contracted rays and the positive adjoint on each flipped side, not absolute nefness of the final adjoint. Since p^*F_U is effective and exceptional over X, the exceptional identity and nefness of (\mu p)^*\alpha give

Pushing forward by q proves the assertion. In particular, the negative part in this statement is absolute, even when the program that produced V was relative.□

Proof. For a prime divisor P\subset S, write m_Q for the multiplicity of Q in g^*P, where Q ranges over the prime divisors dominating P. These multiplicities may be computed over the smooth generic locus of P, where P is Cartier. Put

Only finitely many b_P are nonzero, since any such P is the image of a component of F. Thus F_S is a genuine effective Weil real divisor. Global Weil \mathbb{Q}-factoriality makes this finite divisor real Cartier.

We first check equality over codimension one in S. Work near a general point of P and restrict to a transverse disk. Resolving the source, and cutting by general relative ample hypersurfaces if the relative dimension exceeds one, reduces to a projective surface over that disk with connected fibres. These operations can be performed over a relatively compact Stein neighbourhood; they do not require global sections on S. The intersection matrix of the components of the special fibre is negative semidefinite, with kernel generated by the full fibre with its multiplicities. The restriction of F has zero intersection with every fibre component. Its coefficients are therefore proportional to those multiplicities. Equivalently, all the ratios in the definition of b_P are equal. When g is birational this assertion over the generic point of P is immediate.

Consequently

is a signed real Cartier divisor supported over a subset of codimension at least two in S. Moreover, G is numerically trivial over S. We check that such a signed exceptional divisor is zero. The assertion is local on S. Over a relatively compact Stein neighbourhood, take d=\dim V-\dim S general relative ample hypersurfaces and resolve their intersection. They give a projective generically finite morphism H\to S. The cuts may be chosen to meet any prescribed component of G in a nonzero divisorial trace. After normalization and Stein factorization, H\to S factors as a projective birational morphism H\to S' followed by a finite morphism S'\to S. The restricted divisor G|_H is exceptional for H\to S' and numerically trivial over S'. Applying the ordinary exceptional negativity lemma to both G|_H and -G|_H gives G|_H=0. The choice of the cuts therefore excludes every nonzero component of G. Hence G=0.

The surface argument above is also the usual proof of the degenerate-divisor negativity statement: after subtracting the minimum multiple of the full fibre, an effective residual fibre divisor omits a component and cannot be numerically trivial. Notice that projectivity of g was a hypothesis; it was not deduced from factoriality.□

Proof. Nefness gives N_\sigma(D_S)\leq F_S. Fix a prime divisor P\subset S and a prime divisor Q\subset V dominating it. Write m=\operatorname{mult}_Q(g^*P)>0 and b=\operatorname{coeff}_P F_S. At their generic smooth points, pullback of positive currents satisfies

Indeed, the divisorial term \nu(T,P)[P] pulls back with multiplicity m, and the residual current has zero generic Lelong number there.

Choose Kähler forms \omega_S,\omega_V and a constant C>0 for which C\omega_V-g^*\omega_S is Kähler. For \varepsilon>0, let T_\varepsilon be any positive current in the big class D_S+\varepsilon[\omega_S]. Minimality of the multiplicity and monotonicity under adding a nef class give

Taking the infimum over T_\varepsilon, and then letting \varepsilon\downarrow0, gives

This proves the reverse inequality for every P. All multiplicities are unchanged by resolving away from the generic points in question, so the same argument applies to the normal spaces under consideration.

The \varepsilon perturbation is necessary: at a pseudoeffective boundary class, the infimum over its exact positive currents need not equal its minimal multiplicity. The argument uses that equality only for the big perturbed classes.□

Proof. Write N'=N_\sigma(p^*\xi). Since p^*P is nef, N'\leq p^*F. Birational invariance gives p_*N'=F, so E=p^*F-N'\geq0 is p-exceptional. The positive part of p^*\xi is

and is modified nef. If E\neq0, exceptional negativity in its covering-curve form provides a component of E covered by curves C_t contracted by p, with E\cdot C_t<0 [19, Appendix A, Lemma A.3]. A modified nef class has nonnegative intersection with a general member of a family of curves covering a prime divisor: use currents with arbitrarily small negative part and zero generic divisorial Lelong number, restrict to a general member, and let the negative bound tend to zero. But here

a contradiction. Thus E=0.□

Proof. Nefness descends under the surjective morphism g, so \alpha_S is nef. The class of F_V is pulled back from S, so F_V is numerically trivial over S. It is therefore vertical: an effective horizontal component would restrict to a nonzero effective divisor on a general projective fibre, with positive intersection against a suitable power of an ample class, contradicting numerical triviality. For a birational g, verticality is automatic by dimension. First take a projective resolution \pi:\widetilde V\to V of the main component of V\times_S S^{\mathrm q}. The induced map \widetilde g:\widetilde V\to S^{\mathrm q} is projective and has connected fibres. Pull back \alpha_V,D_V,F_V along \pi, and pull back \alpha_S,D_S along s. All displayed class identities are preserved, and \pi^*F_V remains an effective vertical divisor. Lemma 3.42 gives N_\sigma(D_V)=F_V before this modification. Because D_V=c\alpha_V+[F_V] with \alpha_V nef, Lemma 3.45 then gives

We may therefore replace (V,S,g) by (\widetilde V,S^{\mathrm q},\widetilde g) and suppress the new superscripts in the rest of the proof. The good-model comparison now reads p^*D_S=q^*D_m+[E]. No assertion that the crepant subboundary on \widetilde V is effective is needed: this space is used only for classes, currents and the effective divisor \pi^*F_V. The lower-dimensional generalized pair is the crepant pullback to the small model S^{\mathrm q}.

The class identity [F_V]=g^*(D_S-c\alpha_S) shows that F_V is numerically trivial over S. Lemma 3.43 gives an actual effective real Cartier divisor F_S with F_V=g^*F_S. Injectivity of pullback yields

The established identity N_\sigma(D_V)=F_V and Lemma 3.44 therefore give N_\sigma(D_S)=F_S, and Lemma 3.45 gives

On the other hand q^*D_m is nef, so the exceptional-divisor identity applied to the good-model comparison gives

Thus E=p^*F_S as actual divisors. Subtracting them from the comparison proves the exact class equality

This proves the needed preservation of \alpha_S from the final good model, without an assumption about its intermediate models.

Choose a contraction h:S_m\to T to a normal compact Kähler space and a Kähler class \kappa on T with D_m=h^*\kappa. Taking a resolution of the main component of V\times_S W, and then a common projective resolution with U, produces a projective modification r:R\to X and a holomorphic map \ell:R\to T satisfying

For every curve C in a fibre of r this equality implies \ell(C) is a point, since a Kähler class has positive degree on every nonconstant image curve. The fibres of the projective modification r are connected projective complex spaces, hence are connected by chains of curves. Therefore \ell is constant on each fibre of r. The factorization theorem for a proper map onto a normal space gives a holomorphic map f:X\to T with \ell=f\circ r. Pushing forward the class equality by r gives

The maps from the main fibre-product component to S_m have connected general fibres; their Stein factorizations are finite birational over the normal space S_m, hence have connected fibres everywhere. Together with the connected fibres of h, this shows that \ell, and therefore f, has connected fibres. This is the required semiampleness of \alpha.

Finally, R\to S_m is projective: it is obtained from the projective map g by base change, followed by projective resolutions and the projective map q. If h is Moishezon, choose a projective surjection H\to S_m such that H\to T is projective. A component of R\times_{S_m}H dominating R is projective and surjective over X, and its map to T is projective. This is a Moishezon witness for f.□

Proof. The big non-klt contraction. Apply the construction of [39, Section 3]. On its smooth modification \nu:U\to X it writes

The induction over the jumping coefficients of the multiplier ideal reduces the new reduced stratum to dimension at most d-1. The given contraction on the old non-klt subspace and \mathsf C_{d-1} provide the Moishezon maps of those strata. Every map that is promoted to a projective map in that construction contracts exactly the curves on which \nu^*\alpha vanishes. On each such curve the actual real line bundle -D_U has degree \eta_U\cdot C. On a further projective common resolution q:U''\to U, choose an effective exceptional divisor E with -E relatively ample and choose \varepsilon>0 sufficiently small. Use the decomposition

Here \eta'' is Kähler, and the actual real line bundle -D'' has degree \eta''\cdot C on every contracted curve. Restrict this decomposition to the smooth reduced components in the gluing construction. Lemma 3.20 then supplies precisely the relative ampleness needed in place of [39, Lemma 2.42], including the common-resolution step in its Claim 3.5. The exceptional term -\varepsilon[E] makes the positive part \eta'' Kähler on the common resolution.

The exact sequences of multiplier ideals, relative vanishing, finite pushouts, and extension from sufficiently thickened \operatorname{Supp}D_U in that proof then apply unchanged. They give the birational Moishezon contraction in \mathsf C_{d,\mathrm{big}}. In particular the gluing step uses neither generalized termination nor a projectivity criterion for an undetected ray.

Preparation for both cases of \mathsf B_d. The modified-big perturbation [39, Lemma 2.23] supplies a nef datum that dominates a Kähler class on its carrier. Subtracting a sufficiently small multiple of the pullback of a Kähler class \omega_X still leaves a nef datum on that carrier. Thus \alpha is a gklt adjoint plus \varepsilon\omega_X, and the cone argument of [39, Lemma 2.45] makes \alpha NQC. This preparation applies whether or not \alpha is big.

The nef-and-big case of \mathsf B_d. For a gklt pair the multiplier ideal is the unit ideal, so the preceding big non-klt assertion now applies and gives a birational contraction h:X\to Y. On a projective resolution p:W\to X chosen projective over Y, relative Kawamata–Viehweg vanishing for the effective exceptional divisor -\lfloor B_W\rfloor gives R^ih_*\mathcal O_X=0 for i>0. Thus Y has rational singularities. Proper birational Bott–Chern descent [18, Lemma 8.7] gives \alpha=h^*\gamma. The descended adjoint is gklt, nef and big, and has no trivial curve. The criterion [38, Theorem 4.3] makes \gamma Kähler.

For completeness, the projectivity input in that criterion is restricted to a map S\to Z' between smooth compact Kähler manifolds with rationally connected general fibre: S is the chosen smooth divisor on a resolution and Z' is the resolution of the null-locus component. The smooth-source, smooth-base argument in [15, Theorem 3.1, Step 1] applies. Its subsequent relative program starts with this projective map. Thus this application of the criterion uses only the smooth-source, smooth-base projectivity argument and an already-projective relative program.

The non-big case of \mathsf B_d. Use a projective small factorialization and the modified-big perturbation of the pair. The non-pseudo-effectivity of K_X gives an MRC fibration by [58]. Choose a smooth Kähler MRC base Z and a smooth modification \mu:U\to X such that U\to Z is projective. Only the smooth case of [15, Theorem 3.1, Step 1] is needed: pullback identifies the holomorphic two-forms since the general fibre is rationally connected. Lemma 3.23 identifies any (1,1)-class on U, modulo a class pulled back from Z, with an actual real line bundle. Lemma 3.24 identifies its relative rays with rays of the analytic cone; both statements persist along the projective relative construction.

Write, as in [39, Theorem 4.1, Claim 4.1],

where \Delta_U is klt and F_U is \mu-exceptional. The modified-big replacement is chosen, as in the cited Claim 4.1, so that the nef datum \mathbf M_U on this smooth carrier is Kähler. If a further projective resolution is needed, subtract a sufficiently small effective exceptional divisor from its nef part and add that divisor to the subboundary; this preserves the adjoint and the klt inequalities. Choose a_0 using the uniform bound of Proposition 3.40. It makes the relative D_U(a_0)-program \alpha_U-trivial and preserves that choice through every step. This is a projective relative program from its first step: the nef data are represented by real line bundles modulo Z. Lemma 3.23 applies on every intermediate model over this fixed smooth base. More precisely, put r:U\to Z and choose one actual global real line bundle L with

The bundle L is relatively ample, because \mathbf M_U is Kähler and \alpha_U is nef. On each of a fixed finite collection of relatively compact Stein charts of Z, choose an effective real divisor \Theta representing L, with (U,\Delta_U+\Theta) klt. The representatives can retain a positive relatively ample part in their boundaries. Here the equality K_U+\Delta_U+\Theta\sim_{\R}K_U+\Delta_U+L is an actual real linear equivalence of line bundles; the equality with D_U(a_0) modulo r^*\gamma is separately an equality of Bott–Chern classes.

Fix these ordinary pairs on the initial charts. Their actual linear equivalences and their Bott–Chern comparisons with D_U(a_0) modulo the fixed base class persist separately throughout a global relative program. Proposition 3.25 applies: the source is smooth, its adjoint is represented by an actual global real line modulo Z, its carrier nef part is relatively ample, and D_U(a_0)=(a_0+1)\mu^*\alpha+[F_U] is pseudo-effective. It gives a finite program U\dasharrow V and a good relative endpoint. This is one global program controlled by fixed ordinary chart pairs. Its construction uses global line algebras, not a choice of unrelated local minimal models. Proposition 3.40 makes every step \alpha_U-trivial with the same value a_0.

Here is an explicit ample-model comparison that gives the required descent of \alpha_V. Both D_V(a_0) and \alpha_V are nef over Z. Put a_1=a_0+1 and a_2=a_0+2. Before the program starts, choose fixed ordinary representatives on the finite chart cover for all three relatively ample nef parts \mathbf M_U+a_j\alpha_U, j=0,1,2, on one simultaneous log resolution. The \alpha_U-crepancy of the constructed program makes its endpoint V a weak model for each of these three fixed ordinary adjoints. Their traces D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V are nef. Corollary 3.26 therefore makes

semiample over Z. Their zero curves are exactly the common zero curves of D_V(a_0) and \alpha_V. Their projective ample-model contractions therefore have the same fibres: these fibres are connected by curves, and each contraction is constant on the fibres of the other. Identify the two normal targets, writing the common contraction as g:V\to S. If D_V(a_j)=g^*\lambda_j, with \lambda_j relatively Kähler over Z, subtraction gives

This is an equality of Bott–Chern classes, not merely of curve degrees. Since the initial program is \alpha_U-trivial, it is also D_U(a_1)-negative. From now on put a=a_1 and suppress the argument in D_U(a) and D_V(a).

The maps V\to Z and S\to Z are projective. Moreover g is projective: a relatively ample bundle for V\to Z restricts to an ample bundle on every fibre of g, and hence is g-ample. Each D_U(a_j) is not big globally: pushing a big such class to X would make (a_j+1)\alpha big. If it were big over the smooth MRC base Z, relative-canonical positivity [39, Theorem 2.48], together with the pseudo-effectivity of K_Z, would make it big globally. It is therefore not big over Z. This property persists to its relative semiample model, so its relative canonical morphism has \dim S<d. Apply Proposition 3.30 to g and the whole adjoint D_V. Its generalized data are effective and gklt. The nef datum is represented on the initial carrier by \mathbf M_U+a\alpha_U, a Kähler class; on a common carrier over V its pullback is nef and big. Adding the strict transform of the effective boundary proves that the total boundary on V is globally modified big. The proposition therefore gives

with effective gklt data on S, a globally nef carrier, and globally modified-big total boundary. Independently, \alpha_S is nef by descent of nefness through the surjective projective map g. The lower-dimensional model assertion will be applied to this whole adjoint D_S.

Take a projective small factorialization s:S^{\mathrm q}\to S and resolve the main component of V\times_S S^{\mathrm q}. The resulting maps \pi:\widetilde V\to V and \widetilde g:\widetilde V\to S^{\mathrm q} are projective, and \widetilde V is compact Kähler. Pull back D_V, \alpha_V and F_V to \widetilde V, and D_S and \alpha_S to S^{\mathrm q}. The pair on S^{\mathrm q} is gklt and its boundary-plus-nef class remains modified big. Lemma 3.42 first gives N_\sigma(D_V)=F_V. Lemma 3.45 then gives N_\sigma(\pi^*D_V)=\pi^*F_V. The new total space is used only as a carrier for this equality and for pullbacks of positive currents; no effective boundary on it is needed. Rename these spaces V and S. No additional relative program is needed for this modification.

The vertical-divisor descent and negative-part lemmas above now give actual effective divisors F_S,F_V with

Apply \mathsf M_{d-1} to the generalized pair with adjoint D_S. On a common resolution p:W\to S, q:W\to S_m of the resulting good model, write

The negative-part comparison proved above gives E=p^*F_S, and hence

Since D_m is semiample, \alpha_S is semiample after descent through p. Pulling back by g and using the \alpha-trivial initial program gives semiampleness of \alpha on X. The target is compact Kähler, and the resulting map is Moishezon, as follows by taking common projective modifications of the maps just constructed. The comparison uses only the chosen lower good model; no lower-dimensional program is lifted to V.□

Proof. The local polyhedrality of the negative cone makes R an exposed ray. Choose a nef support \lambda whose null analytic face is R. Normalize the analytic cone by a Kähler class. On a neighbourhood of the point representing R, the class -A is positive. On the remaining compact part of this slice, \lambda has a positive minimum. Consequently, for a sufficiently large b, the class \omega_R=b\lambda-A is Kähler. Thus A+\omega_R=b\lambda is nef with precisely the null face R. The gklt presentation with nef carrier datum increased by the pullback of \omega_R has modified-big total boundary. Apply \mathsf B_d to construct its contraction. Lemma 3.16 makes the same global line -L relatively ample, and Proposition 3.18 constructs the flip when needed. For a divisorial contraction the exceptional-prime negativity and integral line descent give strong factoriality on the target: pull back a rank-one reflexive sheaf, remove its degree by a rational multiple of the exceptional divisor, descend the resulting line, and compare on the common big open. Local ordinary log-Fano replacement gives rational singularities by relative vanishing. The supporting target and projective flip are compact Kähler.□

Proof. On a common projective resolution p:V\to W, q:V\to Y, write p^*A_W=q^*A_Y+[E] with E\geq0 exceptional over Y. Put r=\mu p and D=E-p^*F. Then

The divisor -D is r-nef because A_Y is nef over S. Negativity gives D\geq0, hence E\geq p^*F. Every added prime is therefore exceptional over Y. This proves nonextraction from X and gives its effective exceptional comparison with A_Y. Strictness at any contracted prime of X follows from strictness for the chosen log terminal model of W, since F has coefficient zero there. The same semiample endpoint makes the model good.□

Proof. On a common resolution write the actual comparison p^*A_X=q^*A_Y+[E], where E\geq0 is q-exceptional. There are finitely many primes of X contracted by its marking. Let \mathcal P consist of those with coefficient zero in E. Take a projective log resolution r:W\to Y which contains all these valuations and dominates the fixed nef carrier. In the crepant structure boundary B_W, the coefficient of each member of \mathcal P is its original effective boundary coefficient, hence lies in [0,1). Keep these coefficients unchanged. Increase each other r-exceptional coefficient, if necessary from a negative value, to a number in (\max\{0,b_Q\},1). Keep all nonexceptional coefficients unchanged. The resulting effective klt boundary \Gamma_W satisfies

where F\geq0 has precisely the unwanted exceptional primes as its support. The inherited splitting gives actual real-line representatives for every class modulo Y. Since r is birational, the nef carrier datum is relatively big; the adjoint is relatively represented by F and is pseudo-effective. Apply Proposition 3.25 over Y. Its projective endpoint r':Y'\to Y has an effective relatively nef exceptional trace F', so negativity gives F'=0. A prime outside \operatorname{Supp}F cannot be contracted by an F-negative birational program: on a general curve through its generic point the effective divisor has nonnegative degree. Thus all members of \mathcal P survive, and all unwanted exceptional primes disappear. No new prime is extracted by the program. It follows that r' is crepant and that the marking from X contracts exactly primes with strictly positive original comparison coefficient. It is nonextracting and is therefore a log terminal model. The pullback of the semiample adjoint on Y makes it good. Finally apply Corollary 3.39.□

Proof. After the smooth preparation in [39, Proposition 5.12], write the adjoint as

Here N_+(A) denotes the reduced divisor with support \bigcap_{0<\epsilon\ll1}\operatorname{Supp}N_\sigma(A-\epsilon\omega); this support is constant for sufficiently small positive \epsilon and is independent of the chosen Kähler class \omega [39, Lemma 5.3]. At a nonterminal scaling step,

Consequently \omega_i\cdot R_i>0 and D_i\cdot R_i<0. A component of the actual effective divisor D_i supplies a global rational Cartier detector. The supporting contraction follows from \mathsf B_d, and its projectivity and flip follow from Proposition 3.18. This argument does not require \omega_i to remain nef.

To apply \mathsf B_d to a gdlt presentation, lower its finitely many floor coefficients on the initial smooth carrier and add the same small class to the Kähler nef datum. The adjoint does not change, and the resulting gklt presentation persists through the same negative steps. The original gdlt presentation is retained for adjunction to the floor. Thus the special-termination proof of [39, Lemma 5.8] applies with \mathsf F_{d-1} on its smooth lower-dimensional carrier. Its other inputs are adjunction, discrepancy monotonicity, and the local Cartier-index calculation. The last neighbourhood-isomorphism step uses the evaluation-ideal support argument in the support comparison above: the comparison of the first detected flip has support equal to the full inverse image of its exceptional locus, and later comparisons dominate it. One does not infer disjointness of images merely from a vanishing restriction of an arbitrary exceptional divisor. The theta induction of [39, Proposition 5.11] now gives a weak model. Applying \mathsf B_d to its nef adjoint makes it good, and Lemma 3.50 gives the log terminal model. All its steps are detected or belong to the already-projective relative extraction in that lemma.

For completeness, the negative-part identity needed in this argument is only

not the stronger formula N(p^*\xi)=p^*N(\xi) for an arbitrary pseudo-effective class. The comparison with a nef endpoint identifies its exceptional error with the negative part. The formula for N_+ follows by writing a Kähler form upstairs as p^*\omega-F, with F positive on every exceptional prime, and applying the displayed identity to p^*(\xi-\epsilon\omega)+E+\epsilon F. Pushforward identifies the nonexceptional coefficients; \epsilon F supplies all exceptional primes. These are exactly the uses in the theta argument.

Now put P=f^*\omega_S and suppose A+cP is big for some c. Fix \delta>0 smaller than the \omega_S-degree of every compact integral curve on S and enlarge c so c>4d/\delta. Use the following direct preparation, which keeps the original unshifted globally nef datum. On a smooth projective carrier \nu:W\to X write

with the non-Kähler support property used in the big-class preparation. Here \operatorname{Ex}(\nu) below denotes the reduced divisor formed by all \nu-exceptional primes. Write B_W=B_W^+-B_W^-. For small \epsilon>0 put

The new boundary is klt and M'_W is Kähler. Direct addition gives

Moreover B'\geq b. In every theta branch the cumulative added boundary satisfies 0\leq C\leq1-B', so

Thus the unshifted presentation [K_W+b+C/(1+\epsilon)]+M_W is genuinely gklt with the original globally nef datum. This is the presentation used for the horizontal length bound. No base form is subtracted from an arbitrary newly prepared nef datum.

A horizontal negative ray would have P-degree at least \delta; the bound 0<-A_{\rm unshifted}\cdot C\leq4d would contradict c\delta>4d. Every constructed step is therefore over S. Its projective connected fibres are contracted by f, so f factors through its base and through the flip. The final finite extraction is also over S.

The shifted nef endpoint is semiample by \mathsf B_d. Its zero face contains no horizontal curve: decompose such a curve by the unshifted adjoint’s cone theorem; nefness of the shifted class forces all summands into the zero face, and a summand of positive P-degree again contradicts the length bound. The shifted class is big, so the connected semiampleness map is bimeromorphic. Its connected fibres are Moishezon and are connected by chains of compact curves. The preceding zero-face argument makes the map to S constant on each such curve, hence on every fibre. The analytic factorization lemma therefore factors the map to S through the semiampleness map. Subtracting c\omega_S from the normalized class on its target gives the required relative Kähler class for the unshifted adjoint. The effective resolution error from the preparation is removed by the usual negativity comparison, giving a model of the original data.

The forward traces and exceptional splittings persist by Lemma 3.38. In the last extraction, the newly extracted prime classes move from the old exceptional span into the pullback span; they are actual rational Cartier classes. Both-sign negativity proves directness as before.□

Proof. Take a common projective smooth resolution T\to Y, T\to Y_m and resolve the main component of X\times_Y T, further dominating the prepared nef carrier; denote the resulting smooth space by V. Its morphism h:V\to Y_m is projective. Before constructing a program, verify algebraicity for all classes. The projective RC map V\to T and Lemma 3.23 express every class modulo T by actual real lines. The splitting of the chosen Y_m expresses classes from T, modulo Y_m, by exceptional divisor classes. Hence

Lemma 3.24 identifies the relative cone with the full analytic face. These properties and the exceptional splitting persist along the projective relative steps by Lemma 3.38.

Choose the effective gklt preparation on this final carrier. Its adjoint has the identity

where G_{\rm base} is pulled back from the actual effective base comparison and F is exceptional over X, with positive coefficient on every added prime. Its nef datum is nef and big on the prepared carrier. Modulo Y_m it is the actual real-line class [G]-c_1(K_V)-[B_V]. Thus every hypothesis of Proposition 3.25 holds, with the effective relative adjoint G. Denote its global relative endpoint by V_m and the strict transform of G by G_m.

On a common resolution p:W\to V, q:W\to V_m, the actual adjoint comparison has effective q-exceptional error E. The difference p^*G-q^*G_m-E has zero Bott–Chern class and is q-exceptional, because q_*p^*G=G_m. Negativity in both signs makes it zero:

Choose a big common-isomorphism open U\subset Y_m for Y\dashrightarrow Y_m. Such an open exists because that map is nonextracting. On U we have G_{\rm base}=0. Let r:W_U\to X_U be the induced projective birational map. The divisor q^*G_m is r-nef, since the endpoint adjoint is nef over U and the base class has zero degree on r-fibres. Equation (*) gives

since F is exceptional over X. Negativity yields q^*G_m\leq0. Effectivity yields G_m|_U=0. This step removes the possible horizontal components of F; they were not assumed very exceptional at the start.

Now G_m is supported over codimension at least two in Y_m. It is nef over Y_m, so very-exceptional negativity gives G_m=0. The endpoint adjoint is the pullback of A_{Y_m}, hence is nef and good over S. Its relative canonical morphism is Moishezon: it is the connected factor of the composite of the projective morphism to Y_m and the lower-dimensional Moishezon canonical morphism. Lemma 3.49 removes the added source-exceptional errors and proves nonextraction. All these operations are over S.

The all-class invariant was established before the relative program; its preservation follows at each step from the same cone identification and forward cohomology splitting. If the construction has contracted any original prime crepantly, apply Lemma 3.50 to retain it.□

Proof. First take the usual smooth carrier with effective gklt boundary and adjoint \mu^*A+[F_0], where F_0 is effective exceptional with full exceptional support. Its boundary-plus-nef trace is big and its adjoint is not big, so its canonical class is not pseudo-effective. Take its global MRC and resolve the MRC graph and the fixed carrier, obtaining a projective map to a smooth compact Kähler Z with K_Z pseudo-effective. Perform the Kähler-nef-part preparation after these graph resolutions, preserving the maps to Z and S. This order ensures that the final scaling datum is Kähler, rather than merely the nef pullback of an earlier Kähler datum. Write the resulting adjoint as

The global modified-big hypothesis makes a preliminary base shift unnecessary. Relative pseudo-effectivity is preserved. No base shift of A_0 is big, since pushing a big such class down would make a base shift of A big. The fixed morphism h:U\to Z is projective and has RC general fibres. All these preparations precede the next, fresh base shift.

Put P=f_U^*\omega_S and choose c>2d/\delta, where every integral curve on S has \omega_S-degree greater than \delta>0. Set

Since A_c is not big globally, relative-canonical positivity [39, Theorem 2.48(2)], together with pseudo-effectivity of K_Z, shows that A_c is not big over Z, whether or not it is globally pseudo-effective. The class N is big over Z. Consequently D cannot be pseudo-effective over Z: otherwise D+N=A_c would be big over Z. The smooth projective RC Hodge projection represents every class modulo Z by an actual real line. The ordinary global klt adjoint D is therefore relatively non-pseudo-effective. The projective ordinary Mori construction of Corollary 3.28, with scaling of the actual relatively ample real line represented by N, terminates in a Mori fibre space. Lemma 3.23 and Lemma 3.24 identify every relative ray with a full analytic ray. Its nef b-data retain the fixed globally nef carrier U.

First suppose A_c is not pseudo-effective globally. The retained relative-canonical positivity theorem on h, together with pseudo-effectivity of K_Z, then says that A_c is not pseudo-effective over Z. The terminating relative program ends in a Mori contraction at a threshold \tau>1. At every step its scaled zero ray has A_c-degree negative. Inductively the map to S is retained: if such a ray were horizontal, its P-degree would exceed \delta, while the gklt cone theorem for A_0 gives a rational generator of -A_0-degree at most 2d, contradicting A_c\cdot R\leq0. Once P\cdot R=0, the step is negative or crepant for A_0, so its gklt presentation persists. The same argument applies to the final Mori ray. This would be an A_c-negative Mori contraction over S, contradicting the preserved pseudo-effectivity over S. Hence A_c is pseudo-effective globally.

It is not big globally. Relative-canonical positivity now implies that it is not big over Z. Its pseudo-effective threshold for D+tN over Z is exactly one: for t>1 the class is big, while pseudo-effectivity for t<1 would make A_c big because N is big over Z. The same finite relative program therefore ends in a Mori contraction g:U'\to Y at threshold one. All birational steps, including any steps at threshold one, and the final zero ray are over S by the preceding length argument with A_c\cdot R\leq0. For a zero ray with positive P-degree, A_0 would again have degree less than -c\delta. Thus Y carries a map to S and \dim Y<d. The class A_c has zero degree on every g-contracted curve. This is an ordinary projective log-Fano contraction: rationalize its effective real ordinary boundary, preserving klt and relative antiampleness. Relative vanishing gives rational singularities on Y, which is compact Kähler because it is projective over Z. Lemma 3.15 therefore gives an actual equality A_c=g^*A_Y in Bott–Chern cohomology. This descent precedes the base presentation. Apply Proposition 3.30 to g and the whole adjoint A_c. The negative or crepant steps preserve its effective gklt data. Its nef part is the trace of M_U+cP from the fixed carrier; on a common carrier it is nef and big, so its total boundary is globally modified big. The proposition gives effective gklt data for A_Y with globally nef carrier and globally modified-big total boundary. Relative pseudo-effectivity descends through the projective surjection g, so \mathsf M_{d-1} applies over S, with the base shift retained. For the source data in Lemma 3.52, use Lemma 3.29 to choose a Kähler nef datum on a smooth carrier while keeping A_c fixed. Apply that graph lemma to the chosen lower-dimensional output.

Adjoint-preserving preparation is compatible with the original pair in this comparison. On a common resolution of a nonextracting good output, keep the original nef datum and subtract the effective model comparison divisor from the original structure divisor. The resulting structure divisor has the same adjoint as the output; its trace is effective by nonextraction, and its discrepancies have not decreased. Strictness on contracted original primes is preserved. Thus the prepared presentation supplies a comparison for the original data as well.

Steps at threshold one can be A_c-crepant divisorial contractions. Consequently the composite constructed so far need only be a weak good model of the original prepared adjoint. Apply Lemma 3.50 to extract the finitely many original primes with zero comparison coefficient. This gives the strict log terminal model and preserves the cohomology invariant. The shift changes neither relative comparisons nor relative nefness. Lemma 3.49 removes the full-support preparation error and proves the claimed \mathsf M_d.□

Proof. First prove, by induction on the parameter-polytope dimension, polyhedrality of the effective locus, finite canonical-model chambers, and finitely many chosen terminal models on those chambers, following [39, Theorem 5.15]. The zero-dimensional case uses \mathsf M_d and uniqueness of the canonical model; it does not yet assert finiteness of all weak models.

At a central point choose X\dashrightarrow X_m\to X_c over S. The transport invariant in \mathsf M_d supplies the traces of the whole nearby polytope on X_m. Shrink so the finitely many strict exceptional inequalities remain strict. Add a common pullback from S so that the central canonical class on X_c is Kähler absolutely. On the boundary of the parameter polytope, apply the smaller-parameter induction over X_c; its existence input is \mathsf M_d(X_c), just proved independently. No projectivity of X_m\to X_c or Hodge projection over the possibly singular X_c is used.

The central adjoint is zero over X_c, so the relative effective locus is the radial cone on its boundary effective locus. Apply the cone-length estimate of [39, Claim 5.1] on the finitely chosen gklt terminal models h_i:Y_i\to X_c, all of dimension d. Write A_{u,i} for the boundary-parameter adjoint, nef over X_c, and \psi_i:Y_i\to Z_i for its relative canonical morphism. The cone theorem is not applied to the possibly lower-dimensional canonical targets Z_i.

Let H_i=h_i^*\omega_c and choose \delta>0 below the \omega_c-degree of every compact integral curve in X_c. Fix \bar\lambda<\delta/(\delta+2d) and for 0<\lambda\leq\bar\lambda put

The same cone estimate shows more than nefness: for a sufficiently small \epsilon>0, uniform on the finite list and the chosen radial interval, A_{\lambda,i}-\epsilon H_i is nef. Indeed on a horizontal A_{u,i}-negative rational ray its degree is greater than (1-\lambda-\epsilon)\delta-2d\lambda>0; on the nonnegative part of the cone and on the vertical cone it is nonnegative. It follows for every class \xi\in\overline{\mathrm{NA}}(Y_i) that

In particular its null curves are exactly those contracted by \psi_i. The central data are crepant through these maps over X_c; thus the convex data defining A_{\lambda,i} remain gklt with globally modified-big boundary. Apply \mathsf B_d to A_{\lambda,i}. Its connected semiampleness map and \psi_i both have Moishezon fibres, and each connected such fibre is connected by chains of compact curves. Equality of their contracted curves makes each map constant on the fibres of the other. The analytic factorization lemma identifies the two maps. Thus the selected relative canonical model is also the canonical model over S on this radial interval. This proves local polyhedrality and the finite local list; compactness gives the global finite list of canonical models and chosen terminal models.

Uniform preparation preserving every weak model. Write A_u=[K_X+B_u+\mathbf M_{u,X}] for the adjoints in the parameter polytope. We first record why a weak model of these gklt data extracts no divisor. On a common resolution p:U\to X, q:U\to Y put D=p^*A_u-q^*A_{u,Y}, using the actual structure-boundary comparison. The weak-model discrepancy inequalities give p_*D\geq0, and -D is p-nef because A_{u,Y} is nef over the fixed base. Negativity therefore gives D\geq0. An extracted prime on Y has coefficient one in the weak-model boundary, whereas its coefficient in the structure boundary of the gklt source is less than one; it would give a negative coefficient of D. Hence there is no such prime. Consequently D is q-exceptional and the target is itself gklt.

Fix a parameter u_0. The boundary plus nef part is globally modified big by hypothesis. The modified-big preparation [39, Lemma 2.23] gives, on a fixed projective smooth carrier a:W\to X, an effective klt boundary \Gamma, a Kähler class \gamma_0, and an effective a-exceptional real divisor F such that

Here is the effectivity detail in this use of the preparation. Its new nef datum dominates a Kähler form on a carrier. After resolving the new structure boundary, choose an effective exceptional divisor whose negative is relatively ample and subtract a sufficiently small multiple from the pulled-back datum to make it Kähler. Add the same multiple to the structure boundary, then replace its negative coefficients by zero and, if needed, add small positive coefficients on the remaining exceptional primes. All coefficients stay below one. Only exceptional coefficients have been increased; their increase is the divisor F\geq0 in (3.32).

On a sufficiently small closed polyhedral neighbourhood of u_0 put

These classes remain Kähler by openness, and the same \Gamma and the same F satisfy

Thus this is one affine family of gklt pairs on one smooth carrier, with a fixed effective boundary and Kähler nef data.

Let \psi:X\dashrightarrow Y be any weak model at a parameter u in this neighbourhood, and resolve the induced map W\dashrightarrow Y by r:T\to W, q:T\to Y. Its original comparison is

Since \psi is nonextracting, every a-exceptional prime is exceptional over Y. Hence r^*F is also q-exceptional, and

This proves that W\dashrightarrow Y is a weak model of the prepared pair. In detail, subtract the effective exceptional error on the right from its structure boundary on T and push down to Y. The resulting boundary is precisely the transform of \Gamma, it is effective, the adjoint is the already existing Bott–Chern class A_{u,Y}, and all discrepancies are at least those of the prepared gklt pair. This uses generalized data at the level of currents. On T keep the nef datum r^*\gamma_u, and subtract the displayed effective q-exceptional error from the prepared structure boundary. Pushing forward by q gives the boundary \Gamma_Y and the trace of this fixed nef b-class. Their sum with the canonical current represents the already existing locally exact adjoint A_{u,Y}; the resolution identity defines its structure boundary. The generalized discrepancies have not decreased. This argument neither requires K_Y+\Gamma_Y to be real Cartier nor asserts that the nef trace is separately a Bott–Chern class. It therefore applies to every Kähler weak target, including the adjunction strata used in special termination.

Every weak model occurs in the finite canonical list. Choose a small box in H^{1,1}(W,\mathbb R) which spans that vector space and whose addition to the compact family \{\gamma_u\} stays inside the Kähler cone. For any Y above choose a Kähler class \eta_Y and define \eta_W=r_*q^*\eta_Y. Smooth-source cohomology puts \eta_W in H^{1,1}(W,\mathbb R) and gives an actual r-exceptional real divisor D_\eta with

Its negative is r-nef, so D_\eta\geq0 by exceptional negativity. As W\dashrightarrow Y extracts no divisors, D_\eta is also q-exceptional. Scale \eta_Y by a sufficiently small positive number so that \eta_W lies in the fixed box. Adding this identity to (3.33) yields an effective q-exceptional comparison with target class A_{u,Y}+\eta_Y, which is Kähler over the original base. Thus W\dashrightarrow Y is the canonical model of a member of this one enlarged prepared polytope. Its canonical-model list is finite by the first part of the geography proof. The induced list of maps from X is therefore finite as well. A finite cover of the original compact parameter polytope completes the proof for all weak models.□

Proof. The assertions in dimensions zero and one follow from degrees and factorization of maps of compact curves. Suppose the package holds in dimensions less than d. Lemma 3.47 first proves \mathsf C_{d,\mathrm{big}} and then \mathsf B_d. The former uses \mathsf C_{d-1}; the nonbig part of the latter uses only a chosen \mathsf M_{d-1} model and the explicit negative-part comparison. In particular, no lower program is lifted.

With \mathsf B_d available, Proposition 3.48 constructs detected negative steps in dimension d. Lemma 3.51, using the already known \mathsf F_{d-1}, gives the big-shift case of \mathsf M_d. Lemma 3.53 gives its remaining case, using \mathsf M_{d-1} and the projective graph construction. This proves \mathsf M_d over every proper compact Kähler base, with its chosen-model invariant. Lemma 3.54, whose hypotheses are both \mathsf B_d and \mathsf M_d, now proves \mathsf F_d.

It remains to prove \mathsf C_d when its nef NQC adjoint \alpha is not big. First use Lemma 3.29, preserving the adjoint \alpha and making the nef datum Kähler on a smooth carrier. Its multiplier-ideal inclusion allows the given contraction to restrict to the prepared non-klt closed subspace by Lemma 3.31. Next take the projective dlt preparation in [39, Section 6], keeping its crepant structure data. The working space is klt and globally strongly \mathbb Q-factorial. The retained crepant structure boundary need not be lc; it is effective, including the extracted primes of coefficient at least one. On a common carrier the nef datum is nef and big. Consequently the total boundary trace is modified big, and \alpha-c_1(K_X) is a big Bott–Chern class. The contraction on the non-klt closed subspace transports through this preparation by Lemma 3.32. Choose an NQC expression with positive coefficients and integral multiples of its rational degree-two summands. Corollary 3.37 provides a single number t such that each negative ray of K_X+t\alpha is \alpha-trivial. Choose a Kähler scaling class \omega with K_X+t\alpha+\omega nef. Every selected negative ray is ordinary K_X-negative; the actual rational canonical line is its detector. Proposition 3.48 supplies its projective contraction or flip, and the fixed integral grid persists. The descended \alpha remains nef by projective descent of nefness.

The driving class is non-pseudo-effective throughout, by the corollary. Its initial pseudo-effective scaling threshold is positive; choose a cutoff \varepsilon>0 below that threshold. On the fixed initial carrier the compact segment of globally nef data

has modified-big total boundary. At its scaling parameter each working model is a marked weak model of a member of this segment. Each selected step is negative for the fixed driving adjoint, so its discrepancies do not decrease and increase at a valuation affected by that step. An infinite sequence would repeat a marked model by \mathsf F_d; the cumulative nonzero effective comparison for the fixed adjoint would then contradict equality of that marking. Hence this program terminates. Non-pseudo-effectivity excludes a nef endpoint, leaving an \alpha-trivial ordinary Mori contraction f:X'\to Z.

The structure data of the prepared contraction problem remain crepant throughout this \alpha-trivial program. Their effective traces and globally nef and big carrier datum therefore persist. Lemma 3.32 transports the contraction on the non-klt closed subspace through each birational step. Lemma 3.15 gives

in Bott–Chern cohomology. The class \gamma is nef by projective descent; Lemma 3.35 descends the fixed rational NQC summands, and projective multisections preserve their nonnegative degrees.

Apply Corollary 3.34 to this projective Mori contraction. In its notation the prepared non-klt subspace N_1 has Stein space V(f_*J_1) over Z. If N_1 dominates Z, this Stein space is Z itself and its inherited contraction already supplies the contraction of \gamma. Otherwise the base presentation has multiplier ideal J_Z\supset f_*J_1, so the contraction restricts to V(J_Z) as a closed subspace of V(f_*J_1). Since \dim Z<d, the assertion \mathsf C_{d-1} then gives the contraction of \gamma. Pullback supplies the contraction on X'. Its pullback to projective common resolutions factors through the original space by Lemma 3.31, because the adjoint classes are crepant. This gives the required Moishezon contraction on the original space. This proves \mathsf C_d and completes the induction.□

Proof. Proposition 3.48 constructs each step and Lemma 3.38 supplies the forward traces from the fixed carrier. Each reached model at a scaling parameter is a marked weak model of that parameter’s adjoint. The compact parameter segment satisfies \mathsf F_d, so there are finitely many such markings. Strict increase for the fixed driving adjoint rules out repetition. If that adjoint is not pseudo-effective, the positive pseudo-effective scaling threshold gives a fixed positive cutoff containing every parameter of a continuing program. The finite program cannot end nef, hence ends with the claimed Mori map.□

Proof. Choose one resolution p:\widehat V\to V witnessing Definition 3.1, and write K_{\widehat V}+\widehat B=p^*(K_V+B) using the actual meromorphic identification. Its unit components are the strict transforms \widehat S_i of the finitely many components S_i of S. Every component \widehat Z of an intersection of distinct \widehat S_i’s maps birationally onto a stratum Z, because the map is an isomorphism at its general point. Conversely every stratum is obtained this way. The general SNC locus makes both the component and its index set unique. Compactness makes the collection finite. The empty intersection is \widehat V itself. Write p_Z=p|_{\widehat Z}:\widehat Z\to Z, so p_V=p.

Here are the adjunction and normality details, including the facts used at deeper intersections. For a chosen set of indices, keep their coefficients one and allow each unused coefficient to be either one or 1/2. Write B^{\mathbf e}=B-\sum_i(1-e_i)S_i. Global \Q-factoriality makes this an actual rational-line operation, and

The subtracted divisor is effective. All exceptional coefficients remain below one, and the remaining unit components are exactly the retained strict transforms. Successive SNC adjunction on \widehat Z gives

Here and below such a restriction equality means equality of the meromorphic residue maps in degree q, and hence equality of the actual lines. If two orders of residue differ by an ordering sign, its q-th power is one.

We induct on the number of chosen indices. At each stratum already constructed, we retain normality, effectivity of every weighted different, and the crepant SNC residue comparison on \widehat Z. For this chosen comparison the exceptional coefficients are below one, the unit components are the retained strict intersections, and the full-weight pair is dlt. These assertions hold for Z=V. Lowering unused unit components preserves dlt; lowering all of them makes the effective pair at the current stratum klt. It has rational singularities and is Cohen–Macaulay by the dimension-free floor-connectedness lemma [53, Lemma 2.1]. If just one unused coefficient is retained, the unit divisor of the induced SNC boundary is a disjoint union R of smooth divisors on \widehat Z. The same lemma gives

The fibers are connected, so the images of different connected components of R cannot meet. Each component maps birationally to its image; the displayed equality, factored through finite normalization, makes that image normal. It is compact Kähler by restriction of the local Kähler potentials from V.

For every allowed weight choice, push forward the SNC different to define the rational different on this new stratum. In codimension one, the residue of a local frame of the ambient invertible line identifies its divisorial adjoint with the restricted line. Reflexive extension on the normal stratum gives Equation (4.1) everywhere. Pulling it back agrees with the original SNC residue on a dense open and therefore everywhere, proving crepancy as an actual meromorphic identification.

The different is effective. At a general point of any of its primes, the normal surface-slice calculation of [53, Lemma 6.2], which is stated in every dimension, preserves exactly the residue order and reduces it to adjunction on a normal surface germ with effective boundary and a smooth marked curve. For completeness, on a minimal resolution of that surface, write the crepant boundary as the effective strict transform plus an exceptional divisor E. For each exceptional curve D,

Indeed B^{\rm str}\cdot D\geq0, and adjunction gives K_{\widetilde T}\cdot D=2p_a(D)-2-D^2\geq0; minimality excludes a smooth rational exceptional (-1)-curve. The negative-definite exceptional intersection matrix gives E\geq0: if E=P-N with disjoint nonnegative parts and N\ne0, then E\cdot N=P\cdot N-N^2>0, contrary to the preceding inequalities. The strict transform is finite birational over the smooth marked curve, and hence isomorphic to it. Adjunction to that smooth strict transform thus has nonnegative coefficient. This is the original coefficient by the surface-slice residue calculation.

The full-weight pair is dlt and has the asserted floor. On an SNC model, at the general point of the center of a divisorial valuation v, take normal parameters x_1,\ldots,x_a, giving an unmarked parameter boundary weight zero. The SNC discrepancy inequality is

Every v(x_i)>0. Thus a zero-discrepancy center is an intersection of unit components. Those intersections meet the isomorphism locus, while every exceptional component has coefficient below one. To produce a dlt resolution for the restricted map, principalize its nonisomorphism locus and resolve together with the ordered SNC boundary. That locus contains no entire log canonical center, so the displayed discrepancy inequality makes every new exceptional coefficient strictly below one. This is the dlt resolution criterion. It also identifies the unit primes downstairs with the next incident strata.

Finally let R_{\rm all} be the whole unit union on \widehat Z. Sections on its smooth components which agree on all their SNC intersections glue by the elementary equalizer for the coordinate ideals of an SNC union. Even iterated residues make the equalities independent of the chosen chain. Apply the floor-connectedness lemma to the full-weight pair and this full unit union. It gives (p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}. Projection formula therefore descends the glued section uniquely to the restricted invertible line on the reduced C_Z. The proof with Z=V gives the assertion for S.□

Proof. We use the dimension-free torsion-free restriction lemma [53, Lemma 7.1]. In its notation, for an effective compact Kähler lc pair (T,G), klt away from its reduced floor C, and a connected map f:T\to P to a normal compact base from which an actual positive adjoint multiple is pulled back, it asserts that R^1f_*\mathcal O_T(-C) is torsion-free. The ideal sequence gives

For T=Z and P=Y_Z, if C_Z is vertical, \mathscr Q has proper support and is zero. Serre vanishing for the kernel of \mathcal O_P\to f_*\mathcal O_C, tensored with N_Z^k, extends every restriction in a uniform large-degree tail. If C_Z dominates, the same display says that a section extends as soon as its values are constant on the reduced floor fiber over a dense open of Y_Z: its class in the torsion-free \mathscr Q\otimes N_Z^k then vanishes, and the local lifts glue. They are unique because C_Z dominates Y_Z, so \mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z} is injective. This uses neither reduced special scheme fibers nor base change at their points.

Suppose now that C_Z dominates Y_Z. There is an effective rational Cartier divisor D with \Supp D=\Supp C_Z: restrict to Z the globally \Q-Cartier floor primes of V not containing Z. For small rational \epsilon>0, (Z,B_Z-\epsilon D) is effective klt. Apply Corollary 3.5 to obtain a projective small strong \Q-factorialization \rho:Z_0\to Z, crepant also for the full pair. Write \Delta_0 for its full boundary and D_0=\rho^*D. The full adjoint

is the actual semiample rational line pulled back from Y_Z, and the lowered klt adjoint is J_{\rm low}=P-\epsilon D_0. Put d=\dim Z, choose m>0 with mP Cartier and generated, and take b>4md. Consider

On a general fiber of the map to Y_Z, its class is -\epsilon\{D_0\}. This is the negative of a nonzero effective divisor because the floor dominates Y_Z, so its pairing with a Kähler power is negative. If H were pseudo-effective, a positive current representing it would restrict to a positive current on almost every resolved smooth fiber, contradicting this pairing. Thus H is not pseudo-effective.

Apply Lemma 3.6 to the lowered klt pair and the nef rational line P. Its H-program with Kähler scaling has ordinary J_{\rm low}-negative steps, is finite, and ends in a projective Mori contraction. Every step, including the final Mori ray, is P-trivial. The same actual Cartier line mP descends at each birational step. Consequently its semiample map to the fixed base Y_Z persists on every working model. That map is constant on each connected final Mori fiber and hence factors through the Mori contraction. We obtain

Both maps have connected fibers. The full adjoint on each model is the trace of P. On a common resolution of a birational step, its natural meromorphic canonical comparison is an exceptional divisor with zero class, since the two actual lines descend from the same line on the contraction base. Exceptional negativity applied to both signs makes that divisor zero. Thus the full pair remains crepant, including its meromorphic adjoint identifications. The full transformed pair (T,G) is effective lc and is klt away from its full floor C^+: there it agrees with the transformed lowered klt pair. The program is nonextracting, so the transformed divisor D^+ has \Supp D^+=\Supp C^+. Since the full adjoint is pulled back from Y_Z, the lowered adjoint on a u-fiber is -\epsilon D^+. Hence D^+ is relatively ample over W.

Compare the connected components of a general floor fiber on the original small model and on T. On a common projective log resolution the full crepant subboundary is the same. Its unit union maps to both floors with connected fibers by [53, Lemma 2.1]. Proper closed surjections with connected fibers preserve connected components, also after restricting over a point of Y_Z. Thus the connected components of the two underlying floor fibers correspond.

Every connected component of the floor fiber on T meets one common general Mori fiber. Indeed C^+\to W is surjective because C^+ is the support of the relatively ample D^+. Apply Equation (4.4) to u. Surjectivity of the floor makes \mathcal O_W\to u_*\mathcal O_{C^+} injective; the displayed torsion-free cokernel then makes u_*\mathcal O_{C^+} torsion-free. Let

be the Stein factorization. The finite space \overline C is reduced, and every irreducible component of \overline C dominates W. To see the last implication algebraically, a vertical minimal prime of a finite reduced algebra over a domain would give, by prime avoidance, a nonzero element killed by a nonzero element of the domain. After shrinking Y_Z to a dense open, the fiber-dimension theorem gives every component of \overline C_y dimension \dim W-\dim Y_Z. The fiber W_y is irreducible of that dimension: a resolution of W still has connected fibers over Y_Z, and a general one is smooth and connected and surjects onto W_y. Finiteness now makes each component of \overline C_y dominate W_y. Consequently each connected component of C^+_y meets u^{-1}(w) for the same general w\in W_y.

The reduced general fiber F of u is irreducible by the same resolution argument. If \dim F\geq2, the support of the effective ample Cartier multiple of D^+|_F, whose support is C^+\cap F, is connected. One elementary proof, valid even if F_{\rm red} is not normal, cuts by general hyperplanes to an integral projective surface. If the ample divisor split into disjoint nonzero Cartier parts D_1,D_2, their pullbacks to a resolution would be orthogonal, while the projection formula gives v^*(D_1+D_2)\cdot v^*D_i>0 for both i. Thus both (v^*D_i)^2>0, contradicting the surface Hodge index theorem. If \dim F=1, choose the fiber also off the singular, boundary intersection, and ramification loci. It is a smooth connected curve and

There is a unit point, so F\simeq\PP^1. Its floor has one or two points; if there are two they exhaust the horizontal boundary. In every connected case the preceding common-fiber observation makes the general floor fiber connected, so no comparison is needed. In the remaining case the two unit points represent all its connected components.

Normalize the horizontal floor primes and take their Stein factorizations over W. If there are two primes, each finite Stein map has degree one and is an isomorphism over normal W; their main fiber product gives a proper bimeromorphic comparison. If there is one prime, its normal Stein space is a double cover of W. The exchange on its general fiber extends over the branch locus: the reduced horizontal non-diagonal component of the double fiber product has finite birational projections to the normal cover and hence both projections are isomorphisms. Its graph lifts to the normalized prime. Composing with the program graphs transports the comparison to two original normal floor primes, possibly the same one. All graph projections are projective, and the primes survive birationally because the program extracts no divisors.

This comparison is crepant as a meromorphic residue identity. On a smooth ruled open, order the markings after an étale local cover, choose a base volume \xi, and put the markings at z=0,\infty. A pulled-back q-pluriadjoint frame has the form

Its two residues are a(w)\xi^{\otimes q} and (-1)^q a(w)\xi^{\otimes q}, which agree. This is the two-marking Poincaré-residue comparison of [47, Section 3, Definition 13 and Proposition 14]. On a resolved graph both restricted actual lines are pulled back from Y_Z; their meromorphic embeddings agree on this dense open and hence everywhere, including along exceptional primes. Equal residues therefore give a constant value on the entire general floor fiber. The horizontal consequence of Equation (4.4) extends the section. Normality descends it from the small model. Its restriction is the prescribed section on the original reduced floor, since every original floor prime is covered birationally and the two restrictions agree after that pullback.

It remains to check Equation (4.3). Let \widehat T be a common smooth resolution of the parent and Mori models and let a,b:H\to\widehat T be the two maps from a resolved branch graph. The resolution of the Mori model is an isomorphism near a general entire \PP^1 fiber: its centers have codimension at least two, hence cannot dominate the ruling base. Every holomorphic two-form on \widehat T comes from the base on that ruled open. In fact its vertical one-form terms vanish on \PP^1, and its remaining coefficients are constant on that compact fiber. Thus a^*-b^* kills H^{2,0}, and also H^{0,2} by conjugation. This rational Hodge map on degree two has image of type (1,1). The Lefschetz (1,1) theorem puts its real image in \NS(H)_{\R}. Apply this to the pulled-back parent class c_Z. The link is therefore an arrow as defined above.□

Proof. The center is a dlt stratum by the SNC discrepancy calculation in the proof of Lemma 4.2. Work over the generic SNC part of D, of codimension a, with normal coordinates x_1,\ldots,x_a for the unit divisors. The zero-discrepancy divisorial valuations centered there are exactly the primitive rational monomial rays in \R_{>0}^a. Here is a local verification that also fixes their normalization. The discrepancy inequality in the proof of Lemma 4.2 says on every log-smooth model that the center of a zero-discrepancy valuation is exactly a unit stratum: an additional unmarked normal parameter, or a parameter with coefficient below one, would give positive discrepancy. Blow up that stratum. If the positive normal orders of the valuation are w_1,\ldots,w_b, the chart indexed by a smallest order w_k replaces these by w_k and the positive members of w_i-w_k. The exceptional divisor is again unit, and the center is exactly the new unit stratum. While there is more than one positive order their sum decreases. Eventually the center is the generic point of one unit divisor, where a normalized divisorial valuation has order one. Reversing these ordinary toric blowups proves monomiality and primitive normalization. Conversely this subtraction algorithm for any primitive positive integer vector terminates at order one, and reversing it extracts that monomial valuation. Comparison on a common graph gives uniqueness of the valuation with those weights and of its center on any model.

For a unit component E meeting the inverse image of the generic SNC part of D, put

If the unit components through a unit stratum Q meeting that inverse image are E_1,\ldots,E_b, define

The source carries its standard lattice \Z^b. For the fixed resolution T, these cones subdivide the orthant over its interior. On rational rays, \phi_Q sends a monomial valuation on T to the same normalized valuation on Z, so it is injective on each cone. Interiors of different cones are disjoint, because the center and the weights of a monomial valuation on T are unique. Every rational ray in the orthant occurs: view its valuation on T and use the same SNC description there. There are only finitely many cones over the generic part of D; density of rational rays therefore gives coverage. Taking subsets of unit components gives the faces, and uniqueness of valuations also on faces identifies intersections as common faces. Consequently \sigma_R is a face of an a-dimensional cone \sigma_{R'} with the same downstairs center, for a unit stratum R'\subseteq R.

Let E_1,\ldots,E_a be the unit components through R'. The integer matrix A=(\ord_{E_j}x_i) of this full cone is unimodular. Indeed every positive primitive integral vector in the source cone is the normalized vector of a divisorial valuation, so its image is primitive. If a prime divided \det A, a nonzero kernel vector modulo that prime, lifted to a positive primitive integral vector, would have nonprimitive image. Hence \det A=\pm1. The strata R' and D have the same dimension. We next prove that their generically finite map has degree one. In local coordinates at a general point of R',

where the u_i are units, the y_j are normal coordinates, and z are coordinates along the stratum. The tangential map is generically étale in characteristic zero. Since A^{-1} is integral, replace y_j by \widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}. Then x_i=\prod_j\widetilde y_j^{A_{ij}}, and the map consisting of \widetilde y and the downstairs tangential coordinates is locally biholomorphic at a general étale point of R'\to D. Two distinct points over a general point of D would therefore give two lifts of nearby target points whose normals approach with any fixed positive rate vector in the interior of this cone. Varying the nonzero leading coefficients fills an open set of nearby torus points, so such a target can be chosen in the locus where p is an isomorphism. This contradiction proves that R'\to D is birational. Finally, after taking residue along R', the normal coefficient of \bigwedge_i dx_i/x_i is \det A times that of \bigwedge_jdy_j/y_j; terms differentiating the units vanish in that residue. Since \det A=\pm1, the even pluriresidue removes the sign.□

Proof. The assertions are immediate for d=0, when every floor is empty. For d>0, choose a common log resolution p:T\to Z, p':T\to Z' of the arrow. For a target floor prime D', start with its strict transform R. It is a unit stratum and maps birationally to D'. Its source center p(R) is a stratum by Lemma 4.4. If R\to p(R) is not birational, that lemma replaces R by a unit stratum inside it which maps birationally onto the same source center. The target center may then shrink. If the map to that center is not birational, apply the lemma on the target side, and continue alternately. Each necessary replacement strictly decreases dimension. The process therefore ends with one unit stratum mapping birationally to lower strata D_0,D_0' on the two sides.

The image in the common system base remains unchanged throughout. Indeed the two maps from the resolved graph to the Stein targets are identified by the arrow, and a replacement keeps its entire image on the side then being treated. Initially the image was f_{Z'}(D'). The terminal comparison of D_0 and D_0' is an arrow: restricting the actual meromorphic identification to iterated residues gives crepancy, with the logarithmic determinant sign removed by Lemma 4.4; restricting the line Chern classes in Equation (4.3) gives the same condition modulo \NS_{\R}. Lower invariance thus identifies the transported and assigned residues on D_0'.

This equality determines the entire section on D'. The Stein map of L_{D'}=L_{Z'}|_{D'} is the Stein factor of f_{Z'}|_{D'}, because the line induced on that Stein space is ample; its target is finite over f_{Z'}(D'). The image of D_0' in this target is a closed irreducible subset of full dimension, since it still maps onto f_{Z'}(D'), and therefore is the whole target. Both sections on D' come from it. Equality after restriction to D_0' proves equality on D'.

The same construction shows that the image of every target floor prime in the system base is the image of a lower source stratum, and the reverse statement follows from the inverse arrow. A floor dominates the system base on one side exactly when it does on the other. This proves the last assertion.□

Proof. The top form gives \kappa(U,K_U)\geq0, so the preceding construction applies. The assertion is immediate for a point base. Otherwise use f:U'\to Y above, and write \alpha also for its pullback to U'. For each positive integer k the canonical integral s\mapsto\int_U|s|^{2/k} on H^0(U,kK_U) is positive, continuous, and invariant under bimeromorphic change of variables. Choose j such that N^j is very ample. The powers of g and their inverses are bounded on the corresponding degree j\ell, which defines Y. This linear action is semisimple with eigenvalues of absolute value one. Also |\mu|=1, by integration of \alpha\overline\alpha. Hence the finite measure

is invariant under h; exceptional sets have measure zero here.

Suppose that h has infinite order. In the projective linear group of this degree, the algebraic closure of a power of h is then a positive-dimensional torus T, acting faithfully on Y. Indeed a bounded linear operator is diagonalizable, and the connected algebraic closure of a cyclic diagonal group is a torus. We will obtain a contradiction from any nontrivial one-parameter subgroup of T.

Let Y^\circ be a dense Zariski open in the smooth locus where f is smooth and the relative top form obtained from \alpha is nonzero. Put r=\dim U'-\dim Y. A general fiber F of an Iitaka map has \kappa(F)=0. It has a nonzero holomorphic top form here, so h^0(F,K_F)=1: two such forms would have a nonconstant ratio and give positive Iitaka dimension. If \xi is a local holomorphic volume frame on Y^\circ and \sigma=\alpha/\xi is the relative top form, fiber integration writes the measure as

The coefficient is smooth and strictly positive after this shrink. We include r=0, when the top Hodge line is the one-dimensional degree-zero line.

Set

Invariance of \nu under h makes the smooth positive densities agree on overlaps, extending the density in Equation (4.5) to Y^*. This open is invariant under T. Its complement is the intersection of the closed algebraic sets h^j(Y\setminus Y^\circ), which is closed algebraic by Noetherianity. Its algebraic stabilizer contains the cyclic group generated by h, and therefore contains its algebraic closure. Choose a one-parameter subgroup \C^*\subset T which acts nontrivially, and a general whole orbit in Y^*. The T-invariance of the domain supplies this whole orbit. We now construct a horizontal period metric on Y^* and prove that its restriction to the orbit vanishes.

For a smooth Kähler family a total Kähler class polarizes the primitive real variation in middle cohomology. On a simply connected open of Y^\circ, take the smallest real subvariation of the full middle cohomology containing the line F^r of top forms and its conjugate. It is the intersection of the flat real Hodge subspaces with this property; finite dimensionality reduces the intersection to a finite one. This construction commutes with restriction and continuation: in a flat trivialization the condition of being a Hodge subspace, and of containing the indicated line, is a real analytic equality for the Hodge projections, so equality on one open continues on the simply connected cover. The primitive variation is one such subvariation, so the smallest one lies in primitive middle cohomology. Its polarization is the middle cup pairing with the usual sign, independent of the total Kähler class, because no power of that class occurs in the middle primitive pairing. The horizontal period metric of this polarized real variation is therefore intrinsic.

This possibly degenerate metric extends consistently to Y^*. To compare the metrics on an overlap of two translates of Y^\circ, shrink the overlap further so that a resolution of the fiberwise birational graph is a smooth family over it. The two pullbacks on middle cohomology are flat Hodge embeddings into the cohomology of this common resolution and carry the top lines to the same line. The smallest subvariation there is the pullback of the smallest subvariation on either side: it is contained in both images, and applying the inverse embedding gives the reverse inclusion. Projection formula identifies their middle cup polarizations. Their horizontal metrics therefore agree on this smaller open and, by continuity, on the overlap.

We claim that the period metric restricts to zero on every whole orbit of this one-parameter subgroup in Y^*; constant orbits are immediate. For a nonconstant orbit, pull it back under \exp:\C\to\C^*. The negative upper bound for holomorphic sectional curvature in horizontal directions of a period domain, together with the curvature inequality for a pulled-back curve metric, applies to its local period representations; it holds for real polarizations as well [36, Theorem 9.1]. On a disk of radius R centered at an arbitrary point, translate the center to z=0 and write the pulled-back metric as \chi(z)|dz|^2, continuous and smooth where positive, with curvature -2\chi^{-1}\partial_z\partial_{\bar z}\log\chi\le-\kappa<0. Compare it with

Direct calculation gives \partial_z\partial_{\bar z}\log\chi_R=\kappa\chi_R/2. If the maximum of \chi/\chi_R exceeded one, it would occur where \chi>0 in the interior, since \chi_R diverges at the boundary and \chi is bounded on the closed disk. At that point,

a contradiction. Thus the Ahlfors–Schwarz comparison gives \chi(0)\le4/(\kappa R^2) [2]. Letting R\to\infty forces the metric to vanish on \C. The argument is local and allows zeros of the induced metric, so the local representations just constructed suffice. Consequently their period maps are locally constant along each such orbit. In any holomorphic volume frame the logarithm of the coefficient in Equation (4.5) is harmonic along each orbit: the top Hodge line is flat there, and \sigma is a nonzero holomorphic multiple of a flat generator.

This is incompatible with finite mass. Diagonalize the one-parameter action in projective coordinates, and choose two coordinates nonzero at a general point with different weights, of difference a\ne0. A small smooth transverse slice Q on which their ratio is one gives a holomorphic local biholomorphism onto its image

with fibers of size at most |a|: the ratio at \Phi(z,t) is z^a. The entire \C^* factor is allowed because Y^* is invariant. In the product frame (dz/z)\wedge dt_1\wedge\cdots\wedge dt_{\dim Q}, write \Phi^*\nu with coefficient \rho(z,t)>0. For fixed t, \log\rho(z,t) is harmonic on \C^*. Its circular mean at |z|=e^u is A(t)u+B(t). Jensen’s inequality therefore gives, with an irrelevant positive normalization,

Tonelli’s theorem contradicts the finite mass of \Phi^*\nu, which is at most |a|\nu(Y) by the bounded covering degree. This proves that h has finite order.□

Proof. We use the pure Hodge structure on H^2(Y,\Q) and its rational Beauville–Bogomolov form q_Y, up to positive rational scale. This form is positive on the real symplectic plane and Lorentzian on H^{1,1}(Y,\R); see [5, Lemma 2.1, Definition 5.2, and Lemmas 5.3 and 5.7]. The pullback and (1,1) identifications for these rational singularities, extension of the two-form [44, Corollary 1.8], and Lemma 3.9 give

Indeed that program lemma gives the real (1,1) equality with a direct exceptional sum; the (2,0) and (0,2) summands compare by form extension, and all the summands in the resulting equality are rational.

Let \mathcal E_R be the rational span of the p-exceptional prime classes. Identify H^2(R,\Q)/\mathcal E_R with H^2(Y,\Q) by Equation (4.6), and define the lattice

This amounts to quotienting the integral lattice by its saturated exceptional sublattice. The lattice is independent of a refinement v:R'\to R: integral Gysin satisfies v_*v^*=1, and the smooth degree-two modification formula says that x'-v^*v_*x' is exceptional and that the new exceptional span is v^* of the old one plus the v-exceptional span. A common smooth refinement therefore identifies the quotient lattices from any two resolutions over Y.

The form \eta^e trivializes K_Y. On a resolution R_g of the graph of g, with projections p_1,p_2, the divisors of the two pulled-back volume forms agree. Terminality says that their positive components are exactly the exceptional primes. Thus the exceptional lists for the two projections are identical; in particular g is small in both directions. Let \mathcal E be their common exceptional rational span, and let

Both maps carry \Lambda_Y onto the image of integral graph cohomology in this quotient. Hence G=(\overline p_1^*)^{-1}\overline p_2^* is a rational Hodge automorphism of H^2(Y,\Q) preserving \Lambda_Y.

It also preserves q_Y. The two symplectic forms pull back up to the scalar \lambda, and integration of their top powers gives |\lambda|=1. Use the usual positive normalization of W=(p_1^*\eta\,p_1^*\overline\eta)^{e-1} =(p_2^*\eta\,p_2^*\overline\eta)^{e-1}. For (1,1) classes x,y, the differences between p_1^*Gx and p_2^*x are exceptional for both projections. Expand p_1^*Gx\,p_1^*Gy-p_2^*x\,p_2^*y into two terms, each with one such exceptional factor and one factor pulled back by the appropriate projection. Projection formula makes both pairings with W zero. This is precisely the (1,1) part of the Beauville–Bogomolov formula. The symplectic plane is preserved isometrically because |\lambda|=1, and the Hodge decomposition is orthogonal. Thus G is a q_Y-isometry.

The class c in the statement has q_Y(c)>0. Since \beta is nef over Y, exceptional negativity applied to the divisor E, with \{E\}=p^*c-[\beta], gives E=\sum_i e_iE_i\geq0. Pair with the positive normalization W_R=(p^*\eta\,p^*\overline\eta)^{e-1}. Exceptional classes pair to zero with p^*c by projection formula, and hence for some constant b_e>0

Every term is nonnegative. The first is strictly positive on the common open where \beta is positive and the symplectic form is nondegenerate.

Set H_\Q=H^2(Y,\Q) and D_\Q=H_\Q\cap H^{1,1}(Y,\C). Lefschetz (1,1) on R, together with the line-trace assertion of Lemma 3.9, identifies D_\Q with the rational span of line Chern classes. It is G-invariant because G is a rational Hodge automorphism. The same line-trace assertion applied to the graph congruence gives Gc-c\in D_\R. The line \C\eta survives in H_\C/D_\C; our immediate aim is to prove that all eigenvalues on this quotient have absolute value one. Write \Sigma=(H^{2,0}\oplus H^{0,2})_\R, the positive real two-plane. There are three possibilities for the restriction of the Lorentz form to D_\R.

If D_\R is negative definite, its orthogonal projection c_0 of c to D_\R^\perp is fixed by G and has positive square by Equation (4.7). On D_\R^\perp, the invariant space \Sigma\oplus\R c_0 is positive definite and its complement is negative definite. All eigenvalues there have absolute value one. If D_\R contains a positive vector, it is nondegenerate with one positive direction. Its orthogonal complement is the positive plane \Sigma plus a negative-definite space, giving the same conclusion. In either case H_\R/D_\R\simeq D_\R^\perp.

In the remaining case D_\R is negative semidefinite with radical a rational isotropic line \ell. The lattice action on \ell is \pm1. On \ell^\perp/\ell the form is the positive plane \Sigma plus a negative-definite (1,1) space, and the quotient H_\R/\ell^\perp is dual to \ell. The invariant flag 0\subset\ell\subset\ell^\perp\subset H_\R therefore shows that all eigenvalues on H_\R have absolute value one; this argument also allows unipotent parts. In all cases the eigenvalues on H_\C/D_\C have absolute value one. The rational quotient H_\Q/D_\Q carries the preserved lattice

where the intersection is saturated. Thus \lambda is an algebraic integer and all its algebraic conjugates have absolute value one. Kronecker’s theorem makes it a root of unity.□

Proof. Put V=H^1(T,\Q) and D_\Q=\NS(T)_\Q. For each eigenvalue \theta of A_\C, let V_\theta be its generalized eigenspace and set d_\theta=\dim V_\theta, p_\theta=\dim(V_\theta\cap H^{1,0}(T)). Complex conjugation gives p_\theta+p_{\overline\theta}=d_\theta=d_{\overline\theta}.

For |\theta|\ne1, project c to the block V_\theta\wedge V_{\overline\theta}, with \bigwedge^2V_\theta understood for real \theta. This projection preserves D_{\overline\Q}. Indeed the Chinese-remainder projectors onto the generalized spaces are polynomials in A with algebraic coefficients. Apply them to the two tensor slots separately and then alternate. The resulting maps are algebraic linear combinations of rational Hodge maps on \bigwedge^2V; these preserve rational (1,1) classes, hence D_\Q by Lefschetz (1,1). Separate slot projections avoid any collision of products of eigenvalues in degree two.

On this block A_2=\bigwedge^2 A has the single generalized eigenvalue \theta\overline\theta=|\theta|^2\ne1. The inverse of A_2-1 there is a finite polynomial in its nilpotent part. Project the congruence for c to the block and apply this inverse. It follows that its block c_{\theta,\overline\theta} belongs to D_\C. Positivity says that this block is a perfect tensor between V_\theta and V_{\overline\theta}, pairing their opposite Hodge types: in a basis of H^{1,0} the relevant matrices are positive-definite principal Hermitian blocks. For real \theta the corresponding alternating tensor is nondegenerate.

The intersection of this block with D_\C is defined over \overline\Q. Nondegeneracy is the nonvanishing of a determinant, so it contains a nondegenerate tensor with algebraic coefficients. For any Galois automorphism \gamma, the conjugate tensor remains in D_{\overline\Q}, hence is still of type (1,1) for the given Hodge decomposition, and is perfect between V_{\gamma(\theta)} and V_{\gamma(\overline\theta)}. Opposite-type perfection yields

This does not replace \gamma(\overline\theta) by \overline{\gamma(\theta)}, and assumes no Galois invariance of the Hodge decomposition.

The top-form scalar is the algebraic integer z=\prod_\theta \theta^{p_\theta}=\det(A|H^{1,0}), also when generalized eigenspaces are nontrivial. For any Galois automorphism \delta, reindex by the image roots and pair complex conjugates:

For |\theta|\ne1 the second equality uses Equation (4.8) with \gamma=\delta^{-1}; the other terms vanish. The last equality is unimodularity of the integral action. Kronecker’s theorem now proves the assertion.□

Proof of Proposition 4.7. Apply Lemma 4.8. After a power, g acts trivially on the canonical Iitaka base. Use the modification \pi:U'\to U and the map f:U'\to Y in its construction, and restrict over a general point where the graph and the fibration are smooth and the fiber meets the isomorphism locus of \pi. Dividing \alpha by a local base volume gives a nonzero top form on the smooth compact fiber F, which has \kappa(F)=0; its returning map scales the form by the corresponding power of \mu. A point fiber gives scalar one. For a positive-dimensional fiber, let \omega_U be a Kähler form representing the class in the proposition and put \beta_F=(\pi|_F)^*\omega_U. This is a smooth semipositive form, strictly positive on the dense open where \pi is an isomorphism. Pulling the original graph congruence to U' and restricting to the fiber shows that the two pullbacks of [\beta_F] on a common resolved graph of the returning map of F differ by real line classes. This weaker positivity suffices below.

Since \dim F\leq d<n, Corollary 3.10, using \Ggood_{\dim F}, supplies a terminal compact Kähler minimal model M of F with torsion canonical line. The descended nonzero reflexive top form trivializes K_M itself. Indeed a power of that form, in a trivialization of a multiple of K_M, is a nonzero holomorphic function on the compact connected M, hence a nonzero constant. On a graph resolution of the returning map the two divisors of this volume form agree; terminality makes their positive components exactly the exceptional primes. The returning map on M is therefore small in both directions.

The Kähler Beauville–Bogomolov decomposition [4, Theorem A] supplies a finite cover, étale in codimension one, of M of the form

where T is a torus, the H_i are primitive symplectic factors, and C is the product of the strict Calabi–Yau factors of dimension at least three. One-dimensional factors are elliptic curves and are absorbed into T; two-dimensional factors are placed among the symplectic ones. An absent factor is a point and contributes scalar one. Reflexive forms on these klt spaces can be computed on resolutions by [44, Corollary 1.8]. A power of the small returning map lifts bimeromorphically to P. To justify this, purity makes the restriction of the cover over M_{\rm reg} finite étale [4, Section 3.2, before Lemma 3.7]. This restriction is connected and, after choosing base points, determines a finite-index subgroup of \pi_1(M_{\rm reg}), well-defined up to conjugacy. Remove the codimension-two exceptional sets from the smooth loci. Removing an analytic subset of complex codimension at least two from a manifold does not change its fundamental group. The compact normal connected M is reduced and paracompact, with bounded local tangent dimension. Its analytic pair (M,M_{\rm sing}) therefore has a real analytic embedding [1, Theorem 1] and a compatible locally finite triangulation [50, Section 3, Theorem 1]. Compactness makes the triangulation finite; barycentric subdivision gives the smooth locus finite CW homotopy type, so its fundamental group is finitely generated. There are only finitely many subgroups of the fixed index of the cover. A power preserves the conjugacy class of its covering subgroup, so it lifts on this open, and normalization over the proper graph extends the lift bimeromorphically. We may take further powers throughout.

We explain why the factors can be considered separately. All holomorphic one-forms on P come from T. Their pullbacks show on the smooth open that the torus coordinate of the map depends only on the torus coordinate. Modulo the ideal generated by positive-degree torus forms, the two-forms have basis the symplectic forms \eta_i. Write \dim H_i=2e_i. The highest nonzero power of \sum a_i\eta_i has order \sum_{a_i\ne0}e_i; thus the locus where its full power vanishes is the union of the coordinate hyperplanes. The map permutes these hyperplanes and hence the lines \C\eta_i, matching their nilpotence orders e_i+1. No torus two-form can be added to an image of \eta_i, since wedging that term with the e_i-th power of the corresponding target symplectic form would contradict that nilpotence order. The kernels of the forms now show that each symplectic coordinate depends only on the corresponding symplectic factor. Take a power to remove the permutations.

If C is positive-dimensional, it has no holomorphic form of degree \dim C-1: the form algebra of a strict factor has only degrees zero and its top degree, and no factor has dimension one. In the pullback of the volume form of C, the component with one differential outside C and all the others in C therefore vanishes. The derivative in the C directions has full rank, because the whole map is bimeromorphic and the other coordinates have just been separated. Its cofactors then force the derivative of the C coordinate in outside directions to vanish. Thus that coordinate also depends only on C. These conclusions on a dense smooth open give factor bimeromorphic maps by taking their proper graphs. Their degrees are one because their product has degree one.

The scalar for C is a root of unity by projective pluricanonical finiteness. In detail, H^{2,0} of a smooth compact Kähler resolution of C vanishes by the strict-factor form algebra and reflexive extension. Rational approximation of a Kähler class and Kodaira’s theorem make this resolution projective. Thus C is Moishezon; it is projective by the Kähler projectivity criterion for rational singularities [52, Corollary 6’]. Its canonical line is trivial and its returning comparison is crepant, as seen from the volume form. Chow’s theorem algebraizes its proper analytic graph. The projective log pluricanonical representation theorem [33, Theorem 1.1] gives finite scalar image.

We spell out how the polarizing class reaches the other factors without asserting a cohomology splitting for the singular product P. Choose projective resolutions of its factors with smooth compact Kähler sources, and form the smooth compact Kähler product

Resolve the map P\dashrightarrow F together with X_P\to P. This gives a smooth compact Kähler space R, a modification v:R\to X_P, and a morphism r:R\to F. Put \beta=r^*\beta_F. It is smooth semipositive and strictly positive on a dense open, since r is generically finite. Put a=v_*[\beta]. The smooth modification formula gives [\beta]=v^*a+\{E_v\} for an exceptional real divisor E_v. The graph congruence for [\beta_F] pulls to the cover and, by Gysin pushforward, gives the graph congruence for a modulo real line classes; the exceptional differences are themselves line classes. Pass to a common refinement and use the product of resolutions of the separated factor graphs to compute this congruence; refinements change it only by exceptional line classes. The restriction a_i of a to one factor slice is independent of the other coordinates, by Künneth on the smooth product X_P. Restricting the product graph therefore compares a_i to itself modulo line classes on the factor graph.

Choose the factor slice generally so that it meets the open where \beta is strictly positive and meets each v-exceptional center in codimension at least two, or avoids that center. This is possible by the fiber dimension theorem applied to each center and the projection to the other factors. On the resolved strict transform of this slice, the restriction \beta_i is still semipositive and strictly positive on an open, and the restriction of E_v is exceptional over the factor. Thus the pullback of a_i differs from [\beta_i] only by the class of an exceptional real divisor. This gives both the required positivity and the congruence on each smooth factor model.

For a symplectic factor, its symplectic volume trivializes its canonical line. Canonical singularities preserve the canonical ring on resolution, so a smooth resolution has Kodaira dimension zero. Its dimension is \dim H_i\leq\dim F<n. Corollary 3.10, using \Ggood_{\dim H_i}, runs from that resolution to a terminal model. It is still primitive symplectic: the unique two-form extends, and its top power trivializes the canonical line and is nondegenerate on the smooth locus; irregularity and the form algebra are unchanged. Take a common projective resolution of this terminal model carrying the pullback of \beta_i. Lemma 3.9 decomposes its class as the pullback of a class c on the terminal model plus the class of an exceptional real divisor. The line-trace assertion of that lemma sends the line Chern classes in the factor graph congruence to rational line classes on the terminal model. Pushing through the exceptional quotient therefore shows that the returning graph compares c to itself modulo line classes. These are the hypotheses of Lemma 4.9. That lemma makes the scalar on its two-form, and therefore on its top form, a root of unity.

For the torus, push the semipositive form to the torus as a positive current and average it over translations. The average is a smooth invariant positive Hermitian representative: for every nonzero tangent direction its average is strictly positive because the original form is strictly positive on an open. A bimeromorphic map of a torus is an affine holomorphic automorphism; its linear part preserves the integral H^1 lattice. The comparison modulo \NS_\R and Lemma 4.10 make its top-form scalar a root of unity. The product of the factor scalars is the appropriate power of \mu. It is a root of unity, and hence so is \mu.□

Proof of Proposition 4.6. Put d=\dim Z and m=qk, so sections of L_Z^k are meromorphic m-pluricanonical forms. Let

The integral is computed on the regular locus, or on any resolution by change of variables. This is a vector space: for 2/m\leq1 the elementary power inequality preserves integrability under sums. On a log resolution an adjoint section is locally

It is integrable if h vanishes on every unit component; smaller coefficients cause no obstruction. Every unit valuation has center in C_Z. Thus E_m contains pullbacks of base sections vanishing along f_Z(C_Z). For large k these define Y_Z birationally: multiply one nonzero section of a high ample power vanishing on that proper subset by a complete very ample system. If Y_Z is a point, verticality forces C_Z=\varnothing and a nonzero section is integrable. In particular E_m\ne0.

The function s\mapsto\int|s|^{2/m} is continuous on E_m and positive away from zero. In an integrable basis, the densities of bounded linear combinations are dominated by a constant times the sum of the basis densities, proving continuity by dominated convergence. Its invariance under arrows makes the action and its inverse uniformly bounded in any norm on E_m. Moreover this representation detects the representation on all of H^0(Z,mJ_Z). If an arrow acts identically on E_m, it acts identically on Y_Z because that system is birational. For 0\ne s\in E_m and any other section t, the ratio t/s comes from the rational function field of Y_Z. The arrow fixes both this ratio and s, and hence fixes t.

Take an eigenform s\in E_m\setminus\{0\} for the transport \gamma_* of one self-arrow, with eigenvalue \lambda. In the orientation fixed above, \gamma_*=(\gamma^{-1})^*; put g=\gamma^{-1}, so g^*s=\lambda s. Form the full normalized analytic cyclic root cover of this meromorphic pluriform and take a projective resolution of all its components. Locally, if s=a\eta^{\otimes m} in a meromorphic canonical frame, the cover is the normalization of t^m=a, and its tautological top form is \alpha=t\eta. These descriptions glue when the frame changes. The finite analytic normalization and this full-root construction are also described in the proof of [53, Lemma 7.1]. On the smooth compact Kähler resolution U_s of the full cover, let \pi_s:U_s\to Z be the composite map. Change of variables gives

A meromorphic top form with finite local L^2 norm has no pole, by the one-variable integral transverse to a putative pole. Thus \alpha is holomorphic. Choosing \mu^m=\lambda lifts g bimeromorphically to the full cover, possibly permuting components, with \widetilde g^*\alpha=\mu\alpha.

The lift preserves a Kähler class modulo \NS_\R. A sufficiently large multiple of the pullback of c_Z, plus a relatively ample line class for the finite cover and its projective resolution, is Kähler on U_s. Pulling up Equation (4.3) compares the first summand, and the changes of the relatively ample classes are line classes. After a power fixes one connected component, that component has dimension d=\dim Z<n, so Proposition 4.7 applies there. It follows that \mu, and hence \lambda, is a root of unity.

The possible orders are bounded in this fixed degree m. We give the parameter argument because pointwise torsion alone would not make a bounded group finite. Fix a smooth resolution X of Z and an effective divisor D clearing the poles of a basis of E_m. Put Q_m=\PP(E_m^\vee), the projective parameter space of lines of forms under our quotient convention. The universal section on X\times Q_m belongs to

Pull the parameter space back by the finite coordinate-power map [z_0:\cdots:z_b]\mapsto[z_0^m:\cdots:z_b^m]. Its pullback of \mathcal O(1) is \mathcal O(m), so the displayed line now has an m-th root \mathscr L. The cyclic algebra \bigoplus_{i=0}^{m-1}\mathscr L^{-i}, with multiplication defined by the universal section, defines a finite locally free family \mathcal C of rank m. Every parameter is a nonzero form. Each fiber is therefore reduced: its algebra is torsion-free over the smooth X and is generically squarefree. Every fiber component meets the inverse image W\subset\mathcal C of the nonvanishing locus of the universal section; there the cover is finite étale. Normalize \mathcal C and resolve the normalization projectively. The composite \rho:\widetilde{\mathcal C}\to\mathcal C can be chosen to be an isomorphism over W. Put E_\rho=\rho^{-1}(\mathcal C\setminus W). The locus \{t:\dim(E_\rho)_t\ge d\} is proper analytic by the proper fiber-dimension theorem: every component of the total family meets W, and its general fiber has dimension d. Remove this locus and the critical values of the resolved map. Over the remaining dense open the family is smooth and proper, and each fiber is the disjoint union of smooth resolutions of the corresponding root-cover components, because every component meets the locus where \rho is an isomorphism. Proper smooth transport makes the sum of their middle Betti numbers constant there. Resolve the finitely many irreducible components of the proper analytic complement. On each such resolution pull back the original finite cyclic family, take its reduction, and normalize and resolve anew; restricting the previous normalization is not required. The parameter dimension decreases, so finitely many such steps cover all parameters. There is consequently a uniform bound B_m for the middle Betti number of the entire disjoint union of selected smooth resolutions of every full cover. This does not assert a bound for arbitrary further resolutions.

On the disjoint union of the selected resolutions, let p_1,p_2 be the projections of a smooth resolved graph of the lift. The integral Gysin endomorphism (p_1)_!p_2^* on middle cohomology has \mu as an eigenvalue: the nonzero holomorphic top form has a nonzero cohomology class, p_2^*[\alpha]=\mu p_1^*[\alpha], and (p_1)_!p_1^*=\id. If the order of \mu is a, its cyclotomic minimal polynomial therefore has degree \varphi(a)\leq B_m. There are only finitely many such a. Thus the eigenvalues \lambda=\mu^m on E_m lie in a fixed finite set.

The bounded group on E_m has compact closure in \mathrm{GL}(E_m). Every element of that closure still has its eigenvalues in this finite set, by continuity of characteristic polynomials. On the identity component the characteristic polynomial must be that of the identity. A compact linear group is unitarizable, so an element all of whose eigenvalues are one is the identity. The identity component is trivial; a compact Lie group with this property is finite. The image on E_m, and therefore on the full section space, is finite.□

Proof. At the point strata use the same scalar in the canonical zero-form generator 1. The even residue convention identifies these generators along every path and under every point arrow. This gives the required system in dimension zero; if there are no points the empty system is understood.

Suppose the system has been constructed through dimension d-1. Replace it by powers and their span in a common degree k chosen so that Lemma 4.3 applies on every d-stratum, Proposition 4.6 applies on every vertical d-stratum, and \mathcal I_{f_Z(C_Z)}\otimes N_Z^k is generated for every d-stratum Z. These conditions hold in a common divisible tail: there are finitely many strata, and the last condition is Serre’s theorem. For a d-stratum Z, Lemma 4.2 glues the lower tuple to a section on its whole C_Z. Its residues satisfy the one link in Lemma 4.3, if that link is present, because the lower tuple is invariant. That lemma extends the section to Z.

Let \mathcal P_d(k) be the vector space of all tuples through dimension d whose lower part is in \mathcal V_{d-1}(k) and whose new components restrict to that lower part. Call its elements pre-tuples. Its projection to \mathcal V_{d-1}(k) is surjective, since the finitely many extensions can be chosen independently. This system already generates. To check a point z\in Z, put y=f_Z(z). If y\in f_Z(C_Z), choose a floor point over y and a lower tuple nonzero there. Every extension is the pullback of a section of N_Z^k, so it is nonzero at every point over y, including z. If y\notin f_Z(C_Z), the chosen generation of \mathcal I_{f_Z(C_Z)}\otimes N_Z^k supplies a base section nonzero at y. It pulls back to a section vanishing on C_Z; combine it with the zero lower tuple and zero components on the other new strata. This also treats an empty floor. Surjectivity to the lower system preserves generation at lower points.

We impose invariance in dimension d. For a dominating floor, injectivity of restriction and Lemma 4.5 already make every pre-tuple invariant on that stratum. Verticality is invariant under arrows, so the remaining orbits consist entirely of vertical strata.

For each such orbit choose a representative Z, and let G_Z be the finite isotropy image on H^0(Z,L_Z^k) given by Proposition 4.6. For a pre-tuple with component s_Z, form the norm

Choose a common positive integer a divisible by all |G_Z|, use P_Z(s_Z)^{a/|G_Z|}, and transport it to each member of that orbit. This is independent of the chosen arrow: changing that arrow by an isotropy arrow merely permutes the factors in degree k. No finiteness assertion in degree ak is needed. On strata with dominating floor use s_Z^a, and on all lower strata use the componentwise a-th power of the assigned tuple.

The resulting tuple is compatible. By Lemma 4.5, every transported factor in a norm has exactly the prescribed lower restriction. The norm and its indicated power therefore restrict to the a-th power of that restriction, the same value used on all lower strata and in all other orbits. It is invariant in dimension d by construction, and remains invariant below it.

These tuples still generate. Fix a point z in an orbit member. For each of the finitely many transported factors, choose a point over z on a resolution of its comparison graph. Equality of the pulled-back invertible adjoint lines identifies the fiber values. The factor is nonzero at z exactly when the representative component s_Z is nonzero at the corresponding point of Z. Evaluation at each such point is a nonzero linear functional on the generating space \mathcal P_d(k). A finite union of their proper kernels cannot cover a complex vector space. One pre-tuple makes all factors nonzero and gives a norm nonzero at z. At a lower point a nonzero assigned value stays nonzero after its a-th power. Taking the linear span of all the constructed tuples therefore gives a compatible invariant generating system in degree ak. This completes the induction on d.□

Proof of Theorem 4.1. Take the system in Proposition 4.11. Its components on the prime components of S agree on every common lower stratum, so Lemma 4.2 descends each tuple uniquely to a section of the actual line \mathcal O_V(qkJ)|_S. At any x\in S, choose a prime component through x and a tuple nonzero there. It is the pullback of the descended value in the same invertible line fiber at x, so that value is nonzero. The finite-dimensional span of these descended sections generates at every point of the reduced S. This is the claimed global generation of an actual Cartier multiple.□

Proof of Proposition 2.11, assuming Proposition 5.1. We use the reduced-exceptional boundary after every source modification. The effective discrepancy identity in Proposition 2.7 preserves pseudo-effectivity, and that proposition transfers the resulting decomposition back to the original pair. A dominant meromorphic map to a space in class \mathcal C can be resolved on smooth compact Kähler models; taking Stein factorization gives connected fibers. These operations preserve the positive dimensions of the base and the fiber.

We first suppose a(X)>0. If a(X)=n, then X is Moishezon and Kähler, hence projective, and Proposition 2.10 applies. Otherwise resolve the algebraic reduction as a fibration g:X'\to W to a smooth projective variety of dimension a(X). Put J'=K_{X'}+B' for the modified log adjoint. Its restriction to a very general smooth fiber is pseudo-effective. Indeed, one may choose a sequence of positive currents in c_1(J')+\varepsilon_j[\omega], with analytic singularities and \varepsilon_j\downarrow0, and restrict all of them outside a countable union of proper analytic subsets of the base. By \Ggood_{\dim F}, the restricted adjoint has nonnegative Kodaira dimension.

That dimension must be zero. If it were positive, generic coherent base change would give a divisible q for which \mathcal E=g_*\OO_{X'}(qJ') has a fiber system with a positive-dimensional image on very general fibers. A sufficiently large ample twist of the coherent sheaf \mathcal E is generated at the generic point of the projective W. Evaluation would therefore give two sections of \OO_{X'}(qJ')\otimes g^*\OO_W(A), for one ample A, whose ratio is nonconstant on a general g-fiber. This ratio is a meromorphic function on X'. Every meromorphic function on X' factors through its algebraic reduction, a contradiction. Proposition 5.1 now applies.

Suppose next that a(X)=0, and resolve the fibration in the hypothesis as f:(X',B')\to Y. The base may be replaced by a smooth compact Kähler model, and a(Y)=0, since its meromorphic functions pull back to X'. Write d=\dim X'-\dim Y>0. The same restriction argument and \Ggood_d show that

on very general smooth fibers. If this integer is j<d, the relative Iitaka construction of [57, Lemma 2.6] gives, after the same source modifications, a factorization

with connected fibers, \dim W=\dim Y+j, and \kappa(G,K_G+B''_G)=0 on a very general g-fiber. The cited lemma applies to rational SNC boundaries on compact manifolds in class \mathcal C; its conclusion concerns the restricted log adjoint, and does not require finite generation. A Kähler model of W has algebraic dimension zero because it is dominated by X''. Thus 0<\dim W<n, and Proposition 5.1 applies again.

It remains to exclude the possibility that K_F+B'_F is big. The stable-family comparison below requires a horizontal boundary with coefficients strictly less than one, so we first lower the boundary while keeping the restricted adjoint big. Let B'_{\mathrm{hor}} be the sum of the components of B' dominating Y. There is a rational 0<\lambda<1 such that K_F+\lambda B'_{\mathrm{hor},F} is big for very general F. To justify the uniform choice, exclude at once the generic base-change and image-dimension exceptional sets for all rational \lambda and all divisible degrees. On a fiber outside this countable union, openness of the big cone gives one such \lambda, and a single degree has full-dimensional image. Generic base change makes the same choice work very generally. The boundary \lambda B'_{\mathrm{hor}} is horizontal, rational SNC, and has all coefficients strictly less than one. The stable-family comparison [57, Lemma 5.2] therefore supplies a proper generically finite cover Y'\to Y, a surjection Y'\to R to a smooth projective variety, and a projective stable family \mathcal V\to R with positive-dimensional general fiber, such that the main transform of X' is bimeromorphic to \mathcal V\times_RY'.

The main transform still has algebraic dimension zero. Normalize it and factor its proper generically finite map to X' through the normal Stein space. A meromorphic function descends through the birational part. Lemma 2.8 gives its characteristic polynomial over the finite part, with meromorphic coefficients on X'. Those coefficients are constant because a(X')=0, so irreducibility makes the function constant. Normalization and modifications preserve meromorphic functions. It follows also that a(Y')=0, so the surjection from Y' to the projective R forces R to be a point. But then the main transform is bimeromorphic to the product of Y' with a positive-dimensional projective variety. Rational functions on that factor contradict algebraic dimension zero. This excludes the big case and completes the reduction.□

Proof. The flattening construction of [57, Lemma 2.4] gives the stated reference property, and preserves it after any finite sequence of further base modifications. Its source boundary is the one in [57, Lemma 2.1]. We may resolve the discriminant and the horizontal divisor of the relative generator so that all relevant strata are smooth over the dense smooth log locus. Proposition 2.7 of [57] then gives T and a rational parabolic Hodge line M. A positive multiple of that line is the highest rank-one Hodge step of a complex summand of a pure real-polarizable variation with an integral lattice. Theorem 3.1 of [57] gives the factorization M=p^*P_S and the stated positivity, after a further modification and a positive rational rescaling. These are identities in \Pic\otimes\Q, not merely identities of classes.

Here is the order computation that gives Equations (5.1)–(5.3). Choose a degree m for which g_*\OO_X(m(K_{X/W}+B)) has generic rank one and all boundary denominators are cleared. Its reflexive hull is a line on the smooth W. Let s be a meromorphic relative generating form in a local frame of this line. At the generic point of a prime D, choose a nonvanishing ordinary base volume \omega_W, and set

The relative-times-base identification in this formula is taken in the m-th canonical tensor power. The source is resolved over the generic point of D, so all the primes needed for the divisorial test occur in this minimum. In the notation of the proof of [57, Proposition 2.7], the parabolic order is

Thus the corresponding local frame of T+M has order \alpha_D. Comparing its pullback with the log adjoint form defines A_*, with

Changing the generating form multiplies it by a meromorphic base function. This changes both \alpha_D and \beta_D by its normalized base order and leaves the comparison invariant. The local comparisons therefore give the rational divisor and the line identity globally. Equation (5.5) proves Equation (5.2). Horizontally the comparison is precisely the zero divisor of the fiberwise log section, so it is effective and has the asserted restriction.

The same order computation proves the relative injection. In a divisible degree q, a meromorphic base coefficient \varphi gives a regular base section at D exactly when

Its corresponding upstairs orders at the primes E\to D are

Their simultaneous nonnegativity is equivalent to the preceding inequality. Conversely, an upstairs section restricts to a multiple of the generator on general connected fibers, so its ratio with the generator descends meromorphically to W. The order inequalities make it regular outside codimension two on W, and normality extends it. Consequently

Applying the left exact functor h_* proves Equation (5.3). The reverse global comparison follows from the reference property: its only initially untested poles are on g-contracted primes, all exceptional over X_0; pushing to X_0, extending in codimension two, and pulling back with the reduced-exceptional boundary removes them. This is the global comparison in [57, Proposition 2.7].

Finally consider \tau:W'\to W. The parabolic rational line pulls back under SNC modifications, as in the construction preceding [57, Theorem 3.1]. The coefficient T_D at an old strict transform is unchanged. Indeed the sheaf of regular relative adjoint forms inside the common meromorphic generator line is unchanged over the generic point of D by the log modification formula. Equivalently, the old inequalities above already test all vertical primes there; in SNC coordinates logarithmic pullback preserves those inequalities, so new source exceptional primes impose no stronger one. This keeps \alpha_D unchanged in compatible frames, while parabolic pullback keeps \beta_D unchanged. Over a new \tau-exceptional prime every source prime is exceptional over X_0: a pre-existing one had image of codimension at least two on W, and a new one is exceptional by construction. All therefore have boundary coefficient one. The formula for T_D=\alpha_D-\beta_D then gives coefficient one on that new base prime. Thus

The usual log discrepancy formula for the SNC pair (W,T) now gives K_{W'}+T'=\tau^*(K_W+T)+E_H, with E_H effective and exceptional. Adding the pulled-back parabolic line proves Equation (5.4).□

Proof. On a very general smooth fiber F, use the model supplied by \Ggood_{\dim F} to write \mu_F^*J_F\sim_{\Q}P_F+N(\mu_F^*J_F). Lemma 2.5 identifies the divisible section spaces with those of P_F, so \kappa(P_F)=\kappa(F,J_F)=0; a semiample line of Kodaira dimension zero is torsion. Proposition 2.9, applied to the modification \mu_F with zero added divisor, gives J_F\sim_{\Q}N(J_F) on F, with N(J_F) rational. Lemma 2.5 then identifies the normalized zero divisor of the prepared relative generator with N(J_F). Thus

We compare these fiber multiplicities with those on X. For each of the finitely many components E of A_{*,\mathrm{hor}}, choose a countable sequence of small-Kähler-perturbation currents with analytic singularities whose generic orders at E approach \nu_E(J). Choose a common very general F for these sequences and these components. The currents restrict positively after the same perturbations, and analytic singularities ensure that their orders along the components of E|_F are their generic orders along E. Each restricted order is at least the corresponding perturbed minimal multiplicity on F. Passing to the limit gives (A_*)_E\leq\nu_E(J). This proves

Choose a positive curvature current for the rational line J. Every such current contains its divisorial negative part, hence subtracting [A_{*,\mathrm{hor}}] leaves a positive current. On the inverse image of a dense smooth open W^\circ, with all vertical data removed, Equation (5.1) identifies the resulting singular metric with one on g^*H. In a frame pulled back from W^\circ its weight is plurisubharmonic. It is constant on each compact connected smooth fiber, by the maximum principle (with the value -\infty allowed). Local holomorphic sections of the submersion show that these values form a plurisubharmonic weight on H|_{W^\circ}.

This weight extends across the missing divisors. At the generic point of any such divisor D, choose E\to D for which (A_*)_E=0, using Equation (5.2). At a general point of E away from the other comparison divisors, the comparison after subtracting the horizontal part has neither a zero nor a pole. We can choose local coordinates, with the remaining vertical coordinates denoted by w, in which

Indeed g|_E has maximal tangential rank generically and a local equation of g^*D is a unit times u^{a_E}; the unit has a local a_E-th root. A smaller such chart covers a neighborhood of the base point. The upstairs plurisubharmonic weight is locally bounded above there, and the comparison unit is bounded. Hence the descended weight is locally bounded above near the generic point of D. The removable singularity theorem for plurisubharmonic functions extends it across D away from codimension two, and the Hartogs extension for plurisubharmonic functions extends it across the remaining analytic subset. The extensions respect the line transitions because they agree on the dense open. They give a positive singular metric on H, which proves pseudo-effectivity.□

Proof. The projective quotient S in Lemma 5.2 is a point: otherwise its rational functions pull back nontrivially to W. Thus M\sim_{\Q}0 and H\sim_{\Q}K_W+T. By Lemma 5.3 and \Ggood_{\dim W}, its pullback to a smooth higher model is a semiample rational line plus its negative divisor. Every semiample line on a space of algebraic dimension zero is torsion, since its generated system has zero-dimensional image. Proposition 2.9, applied to this modification with zero added divisor, therefore gives Equation (5.7), including rationality of the divisor.□

Proof. We use the construction in the proof of [56, Proposition 3.3]. Its absolute input is a nef-plus- effective rational decomposition for the canonical divisor of a smooth projective variety with pseudo-effective canonical class. This is supplied here by Proposition 2.10 and its common-resolution comparison.

Replace the image of f by its normal Stein factor. This does not change relative numerical classes or contracted curves, and now f is surjective with connected fibers. Its very general fiber is smooth projective with pseudo-effective canonical divisor, hence is non-uniruled by [8, Theorem 0.2 and Corollary 0.3]. Compactify the projective morphism and resolve away from V. We obtain \bar f:\bar V\to\bar Z, with \bar V smooth projective and \bar Z normal projective, unchanged over Z. If \dim Z=0, V is already projective and the absolute input proves the lemma.

Suppose \dim Z>0, and let r:\widetilde Z\to\bar Z be a smooth projective resolution. Choose a sufficiently positive very ample divisor A_Z on \bar Z and a general B\in|2A_Z|. The pulled-back systems on both \widetilde Z and \bar V are base point free. Bertini makes r^*B and \bar f^*B smooth nonempty reduced divisors. Their double covers

are smooth integral projective varieties. The canonical formulas are

The first line is big when A_Z is sufficiently positive. The rational map V^\sharp\dashrightarrow Z^\sharp has the same very general fibers as f. A covering family of rational curves on V^\sharp would either dominate a covering family of rational curves on Z^\sharp, or cover its very general fibers by vertical rational curves. Both are impossible. Thus V^\sharp is non-uniruled and K_{V^\sharp} is pseudo-effective by BDPP.

Apply the absolute decomposition to V^\sharp. Moving the rational principal difference in the canonical formula into the nef part gives an actual rational divisor decomposition for a pullback of \pi^*(K_{\bar V}+\bar f^*A_Z). Take a common equivariant resolution of that model and its conjugate under the covering involution. On this smooth projective resolution g:T\to V^\sharp, average the two decompositions. We obtain

where \bar P and \bar N are invariant rational Cartier divisors.

Let q:T\to\bar Y be the finite quotient by the involution. The invariant composite \pi g induces a projective birational morphism \bar h:\bar Y\to\bar V, with \bar Y normal and projective. An invariant rational divisor descends by dividing its coefficient at an upstairs prime by the ramification index. The descended divisor is rational Cartier when the original divisor is: after clearing denominators, choose one Cartier equation on a semilocal neighborhood of the finite orbit above a point (a Cartier divisor is principal on a semilocal ring), and multiply all its translates. The product is invariant and has divisor equal to the group order times the original divisor. It descends to a local equation for a multiple of the downstairs divisor. This argument includes stabilizers and ramification primes.

Accordingly \bar P=q^*P and \bar N=q^*N for rational Cartier divisors on \bar Y. Lifting curves through the finite map shows that P is nef, and coefficientwise descent shows that N is effective. Finite pullback is injective on rational divisors, so the displayed identity descends. Restrict it to Y=\bar h^{-1}(V). The term pulled back from A_Z is numerically trivial over Z, giving h^*K_V\equiv_ZP+N, as required.□

Proof of Proposition 5.5. Lemma 5.6 and [64, Theorem 5.2], applied to the generalized pair (U/Z,0+0) with U smooth, prove relative minimal-model existence for smooth varieties by dimension induction. The hypothesis of that theorem is relative smooth minimal-model existence one dimension lower; the lemma supplies its other hypothesis, an NQC weak Zariski decomposition. The induction begins in dimension zero. Then [64, Theorem 5.4] supplies minimal models for every pseudo-effective NQC generalized lc pair in the relevant dimension. Their Theorem 2.7 gives existence in the Birkar–Shokurov sense.

The nef data in the statement are NQC, because a nef rational Cartier divisor is a positive multiple of a nef Cartier divisor. In the pseudo-effective case the preceding paragraph therefore gives the Birkar–Shokurov model required by [64, Theorem 4.2]. In the other case, the non-pseudo-effective alternative of that theorem applies directly. To meet its scaling hypothesis, take an effective sufficiently positive ample rational divisor A whose general components have sufficiently small coefficients. On a fixed log resolution carrying \boldsymbol M, these components are transverse to the boundary and preserve generalized log canonicity, while their ample class can be chosen large enough that D+A is nef. The theorem supplies a terminating MMP starting on V.

We verify the step types inductively. Suppose V_i is \Q-factorial generalized dlt. By [64, Remark 2.15], each birational step has the form

where g_i contracts a negative extremal ray and h_i is small and projective. Suppose g_i is divisorial, and write R for its ray. An exceptional prime E has E\cdot R<0: otherwise it would be effective, exceptional, and g_i-nef, contrary to negativity. There is only one exceptional prime. Indeed, a combination of two distinct exceptional primes can be chosen to have degree zero on R; negativity applied to both signs of that combination would make it zero.

We show directly that Z_i is \Q-factorial. For a prime divisor T on Z_i, let \widetilde T be its strict transform, choose a curve C spanning R, and put

Choose a positive integer m for which m(\widetilde T+aE) is Cartier. Its line has degree zero on R, so [66, Theorem 1.5] gives an actual Cartier line \mathcal L_T on Z_i and an isomorphism

On an open set trivializing \mathcal L_T, the Cartier divisor m(\widetilde T+aE) is principal upstairs. Pushing this principal divisor identity down through the birational g_i gives a principal divisor mT on that open set. Thus T is rational Cartier. It follows that h_i is an isomorphism: the pushforward of an h_i-ample Cartier divisor is rational Cartier on Z_i, and its pullback is the original divisor because h_i is small. It has degree zero on every contracted curve, so relative ampleness rules out positive-dimensional fibers; a finite birational map to the normal Z_i is an isomorphism. If g_i is small, the relatively ample canonical model h_i is the flip of [40, Definition 3.2]. Thus the chosen birational steps are divisorial contractions and flips. The underlying variety of a generalized dlt pair is klt, and these steps preserve the \Q-factorial generalized dlt category [40, Remark 2.3 and Lemma 3.7]. They introduce no prime divisors on the new models. For completeness, on a common resolution of one step V_i\dashrightarrow V_{i+1} carrying \boldsymbol M, taking the difference of the two generalized discrepancy formulas cancels their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new model and anti-nef over it. The negativity lemma makes this difference effective. Composing these actual rational divisor comparisons gives u^*D=v^*D_m+F with F effective and exceptional over the endpoint. The endpoints in the two cases are the nef and Mori fiber endpoints supplied by the terminating program theorem.□

Proof. The smooth pair with boundary T and nef data \boldsymbol M determined by M on W is a rational \Q-factorial generalized dlt pair. Its adjoint H is pseudo-effective by Lemma 5.3. Proposition 5.5 gives a chosen terminating program with scaling to a \Q-factorial generalized dlt model W_{\mathrm{nef}} with nef transformed adjoint

Here M_{\mathrm{nef}} is the trace of the nef b-divisor whose nef representative is M on W. On a common smooth resolution \mu:W'\to W, \sigma:W'\to W_{\mathrm{nef}}, the actual comparison is

The program extracts no divisors. We use this particular model also when the nef data are rationally trivial.

If \kappa(H)>0, Equation (5.9) preserves the divisible section spaces. For \kappa(H)=k the nef H_{\mathrm{nef}} is big, and the second alternative holds with W_m=W_{\mathrm{nef}} and H_m=H_{\mathrm{nef}}. If 0<\kappa(H)<k, the exact global comparison in Lemma 5.2 gives \kappa(J)=\kappa(H). Take the Iitaka fibration of J on the source. Its projective image, followed by Stein factorization and a projective resolution, has dimension \kappa(H). The log adjoint on very general fibers has Kodaira dimension zero by the relative Iitaka construction [57, Lemma 2.6], including its persistence under the reduced-exceptional source convention. This is the first alternative.

Suppose \kappa(H)\leq0 and M\sim_{\Q}0. We may choose zero nef data in this rational equivalence class. The same generalized dlt model W_{\mathrm{nef}} is then an ordinary dlt model and H_{\mathrm{nef}}\sim_{\Q}K_{W_{\mathrm{nef}}}+T_{\mathrm{nef}}. Its adjoint is nef, so Proposition 2.10 makes H_{\mathrm{nef}} semiample as an actual rational line. Since \kappa(H_{\mathrm{nef}})=\kappa(H)\leq0, a generated multiple has constant image and is trivial. Thus the second alternative holds with W_m=W_{\mathrm{nef}} and H_m=H_{\mathrm{nef}}\sim_{\Q}0.

It remains to suppose

Here \dim S>0. For a rational c\in(0,1), put H_c=K_W+T+cM. We claim

Suppose otherwise for one c, and choose an ample rational line U on S. We will find \delta>0 for which H-\delta p^*U has an effective representative, forcing \kappa(H)\geq\dim S>0. Choose a rational a>\max\{1,a_0\} and a small rational \lambda>0 such that K_S+aP_S-\lambda U is big. This is possible because P_S is nef and K_S+a_0P_S is big. Choose

The rational line cP_S+\eta U is ample. A sufficiently high general member divided by its degree pulls back to a rational boundary transverse to T with subunit coefficient. Applying the nonvanishing part of Proposition 2.10 to this ordinary projective log pair gives

A nonzero section defining E_c restricts nontrivially to very general p-fibers in a fixed divisible degree. The hypotheses of the weak-effectivity statement [57, Lemma 3.3] are now satisfied: the source is smooth in class \mathcal C, T is rational SNC, the base S is smooth projective, and a relative log system is nonzero. Applied to the big line K_S+aP_S-\lambda U, it gives

For

the rational interpolation is

Multiplying pullbacks of sections of U by a section of the effective left-hand side yields \kappa(H)\geq\kappa(p^*U)=\dim S>0. This contradicts Equation (5.10) and proves Equation (5.11).

We next run a program to fiber type while descending the actual nef line H_{\mathrm{nef}} through every contraction. Choose the common resolution in Equation (5.9) to determine the nef data. The negativity lemma gives

Indeed its negative is \sigma-nef and exceptional. For every rational 0<e<1, Equation (5.9) gives

If H_{\mathrm{nef}}-eM_{\mathrm{nef}} were pseudo-effective, this equality and effectivity of the last two terms would make \mu^*H_{1-e}, and hence H_{1-e}, pseudo-effective, contrary to Equation (5.11).

Fix rational 0<\varepsilon<1, choose an integer r>0 with rH_{\mathrm{nef}} Cartier, and choose an integer d_0>2kr. This coefficient will force every ray of the second program to have zero degree for the descended nef line. Decreasing the nef data to (1-\varepsilon)\boldsymbol M preserves generalized dlt singularities: on the displayed resolution the crepant boundary changes by -\varepsilon F_M, so discrepancies do not decrease. Add d_0H_{\mathrm{nef}}, determined as nef data on W_{\mathrm{nef}}. It changes no discrepancies, and the resulting adjoint is

It is not pseudo-effective by the preceding paragraph. The non-pseudo-effective case of Proposition 5.5 therefore gives a chosen terminating program to a Mori fiber contraction in the \Q-factorial generalized dlt category.

At any stage of this second program, let H_i be the descended rational line, assuming inductively that it is nef and rH_i is Cartier, and write

An extremal ray negative for G_i+d_0H_i is negative for G_i. The pair with adjoint G_i is still generalized dlt: the added nef b-divisor d_0H_i descends to the current model, so removing it changes no discrepancy. The generalized length bound [40, Proposition 3.13] gives a curve C on that ray with 0<-G_i\cdot C\leq2k. If H_i\cdot C>0, Cartier integrality would give

a contradiction. Thus H_i\cdot C=0. The exact contraction statement [66, Theorem 1.5] descends the Cartier line rH_i along this contraction as an actual Cartier line. In a flip we pull that same line to the flipped side. The descended line is nef: every curve on the contraction base has a curve above it mapping with positive degree, so its degree is nonnegative; pullback preserves nefness. Consequently r and nefness persist, and the argument applies inductively to all steps and to the final contraction. Each step is crepant for H_i.

Let W_m be the last birational model of this second program and let H_m be the descended line on it. Crepancy for the H_i, together with the absence of extraction, turns Equation (5.9) into Equation (5.8) on a common resolution, with an effective divisor exceptional over W_m. The final contraction h:W_m\to V has connected fibers and, by the same exact descent,

for a rational line H_V on the normal projective V. If V is a point, the second alternative holds with H_m\sim_{\Q}0.

Assume \dim V>0. To obtain the first alternative, it remains to show that the log adjoint on the new very general fibers has Kodaira dimension zero. Resolve the base program by \mu:W''\to W and \nu:W''\to W_m, and prepare the source over W'', calling it g'':(X'',B'')\to W''. The diagram we use is

with composite f= h\nu g''. Equation (5.4) and Equation (5.8) give

To see the exceptionality, the error in Equation (5.8) is already exceptional over W_m, and every \mu-exceptional prime is exceptional over W_m as well: otherwise W_m would extract a divisor over W. For every degree divisible by one fixed integer, Equation (5.3), exceptional descent, and projection formula now yield

Here \nu_*\OO_{W''}(qE'')=\OO_{W_m} by normality and h_*\OO_{W_m}=\OO_V. Generic coherent base change, simultaneously for the countably many divisible degrees, bounds the dimension of every such system on a very general f-fiber by one. The log adjoint on that fiber is pseudo-effective by restriction of a countable sequence of perturbed currents with analytic singularities. Its dimension is less than n, so the lower-dimensional \Ggood supplies a nonzero section in some divisible degree. Taking a common multiple shows that the fiber log Kodaira dimension is zero. All maps in the composite have connected fibers; resolving V and the source keeps this property and gives a smooth projective base of dimension 0<\dim V<k. This is the first alternative.□

Proof. Use the chosen base identity in Equation (5.1). In a rationally trivial case fix a rational trivialization of the positive line and put P_0=0; in the big case put P_0=(\nu g)^*H_m. Define the rational divisor

The composed actual line identities give J\sim_{\Q}P_0+A. The divisor A may at first have signed coefficients. We first prove A\geq0.

In a rationally trivial case a divisible multiple of the base line has a section whose normalized zero divisor is exactly E_W. Its section under the exact global comparison has normalized zero divisor A. Regularity of this section proves A\geq0, also on the initially untested primes. In the big case choose a Kodaira decomposition

For rational 0<t<1,

Fix a prime \Gamma\subset X. A sufficiently divisible section of the ample summand can be chosen not to vanish at the generic point of (\nu g)(\Gamma); its pullback has zero order at \Gamma. Multiplication by the section of tG_0 and then the global comparison gives a regular upstairs section. Its normalized order is

Letting t\downarrow0 proves A_\Gamma\geq0. This applies to every prime in A.

Nefness of P_0 now implies N(J)\leq A. Set Q=A-N(J)\geq0, an effective real divisor. By Equation (5.6), Q is vertical. Moreover

is modified nef. We prove Q=0.

Let b:X\to Y denote g:X\to W in the first case and \nu g:X\to W_m in the second. Thus \dim Y=k, and b has connected fibers. If Q\ne0, let

Choose a Kähler class \eta on Y (an ample class when Y is projective) and a Kähler form \omega on X. On H^{1,1}(X,\R) define

In this pairing a divisor or rational line denotes its cohomology class. The exponent is nonnegative since l\leq k-1\leq n-2. The restriction of a modified-nef class to a resolution of every prime is pseudo-effective. Applying this to \{P_0+Q\} and pairing on each component of Q gives \langle P_0+Q,\Gamma\rangle_l\geq0. The P_0-term vanishes by dimension, since it is a pullback from Y and \dim b(\Gamma)\leq l. Summing with the coefficients of Q gives

We now separate images of codimension at least two, where the mixed Hodge index theorem gives strict negativity, from divisorial images, where only a multiple of the whole fiber can have square zero.

If l<k-1, the mixed Hodge index theorem contradicts this inequality. In this form b^*\eta has positive square because l+2\leq k, while \langle Q,b^*\eta\rangle_l=0 by image dimension. The form is a limit of the mixed Hodge index forms obtained by replacing b^*\eta with b^*\eta+\epsilon[\omega]; it therefore has at most one positive direction. In the orthogonal complement of a positive-square vector it is negative semidefinite, with equality only in the radical of the whole form. But \langle Q,\omega\rangle_l>0, since at least one component of Q has image dimension l. Thus Q is not in the radical, and its square is strictly negative, contrary to Equation (5.14).

It remains to treat l=k-1. Terms from primes whose image has smaller dimension vanish in this form, as do cross terms from different divisorial images. Fix a prime divisor D\subset Y and take all the primes E_1,\ldots,E_s\subset X dominating it. Write a_i=\ord_{E_i}(b^*D)>0 and C_{ij}=\langle E_i,E_j\rangle_{k-1}. The divisor D is rational Cartier, also when Y=W_m. Consequently

The first assertion is positivity of the proper intersection of distinct effective primes. The second pairs E_i with the pulled-back divisor D; the additional base factor makes the intersection vanish by dimension. Components of b^*D mapping into a proper subset of D make no contribution. The graph with edges C_{ij}>0 is connected. Indeed over a very general point of D the fiber is connected, all its points lie in the components E_i, and an intersection contributes a positive entry precisely when it dominates D.

For any real vector x=(x_i), the two displayed properties give the exact identity

Thus the block is negative semidefinite with kernel spanned by (a_i). Equation (5.14) forces every nonzero block of Q to equal a positive scalar multiple of (a_i). In particular Q then contains every prime over that divisorial image D.

In the second case this is impossible. The morphism \nu:W\to W_m is an isomorphism over the generic point of D, and E_W is exceptional. Equation (5.2) therefore supplies a component over D whose A-coefficient is zero. Its Q-coefficient is zero as well, contradicting the preceding conclusion.

In the first case, the same zero minimum shows that each divisorial image of a nonzero block of Q is a component of E_W=N(H). Here \{Q\} itself is modified nef. The class

is also modified nef on the smooth W. To see this directly, choose small-perturbation positive currents for \{Q\} with zero generic divisorial Lelong numbers, wedge them with \omega^{n-k}, and push forward. Such a current has no trace mass on a divisor, so its push has no mass on a base divisor: the inverse image is a union of divisors and subsets of higher codimension, all of zero trace mass. The pushed currents therefore have zero generic divisorial Lelong numbers. Their classes tend to the displayed class; a fixed smooth representative for the vanishing error converts this into the defining small-Kähler-perturbation condition for modified nefness. On the other hand, pushing the actual divisor Q in this formula kills the components with smaller image and gives a nonzero positive linear combination of the components of N(H). Boucksom’s exceptionality of the components of a divisorial negative part says that no such combination is modified nef [7, Definition 3.10 and Theorem 3.12]. This final contradiction proves Q=0, and hence Equation (5.13).□

Proof. Put D_Z=\lfloor\Delta_Z\rfloor. We first prove the following assertion for every pair and diagram satisfying the hypotheses of this proposition:

Here \mathbf B(L) is the intersection of the base loci of all positive Cartier multiples of L. Once this is proved, a log canonical threshold at any remaining stable base locus will create a new floor there and give a contradiction.

Take a higher log resolution if needed and put

The equality defining F uses the canonical meromorphic identification. The divisor F is an SNC subboundary. The integral divisor E is effective, r-exceptional, and has no component in common with C: a nonexceptional coefficient of F is a coefficient of the effective \Delta_Z, while every coefficient of F is at most one. The image of C is contained in D_Z. By Theorem 4.1, a divisible multiple of L|_{D_Z} is globally generated on the whole reduced floor.

Suppose first that a component C_i dominates Y. Pulling the floor sections to C_i shows that (f|_{C_i})^*H_Y\sim_{\Q}\ell|_{C_i} is semiample. Semiampleness descends under a proper surjection to a normal space in this situation. Factor C_i\to Y through its normal Stein factor Y_i. Projection formula descends generation through the connected-fiber map. For the finite map Y_i\to Y, at each y\in Y choose a section of the generated pullback line avoiding every point of the finite fiber; finitely many evaluation kernels cannot cover the section space. Lemma 2.8 gives a norm section of a fixed power of the line on Y, nonzero at y. These sections generate a fixed multiple of H_Y. Equation (5.15) and normal birational descent then give semiampleness of L.

We may therefore assume that C is vertical over Y. Choose a sufficiently divisible integer u>1 for which uL|_{D_Z} is generated, and consider the integral line

Although the expression on the right uses rational divisors, its sum is the integral line on the left. The restriction sequence is

The direct image of the last sheaf is supported on the proper analytic subset f(C), so it is torsion. Consequently, the two assertions

will give surjectivity on global sections: torsion-freeness makes the connecting map to R^1f_*\mathscr N zero, and the H^1-vanishing then removes the obstruction to lifting a global section.

We prove Equation (5.18) using analytic injectivity. Equation (5.15) identifies \mathscr N\otimes K_U^{-1} with \Delta_F+(u-1)f^*H_Y in \Pic(U)\otimes\Q; this comparison need not be an isomorphism of the corresponding integral lines. Fix a very ample line A_Y on Y and its Fubini–Study metric. A Kodaira decomposition of the big nef H_Y, given arbitrarily small weight as in the proof of Lemma 5.8, writes (u-1)H_Y as a positive rational multiple of A_Y plus an effective rational divisor whose fixed singular contribution is arbitrarily small and whose remaining contribution is a general divided ample member. Choose the small weight below the log canonical threshold of its pullback relative to the klt SNC boundary \Delta_F, and choose the ample member generally. Then the resulting boundary on U is klt. Choose a positive integer q_0 clearing all its denominators and realizing both the chosen Kodaira decomposition and the preceding rational-line comparison as line isomorphisms, in particular

The divisor metrics and the Fubini–Study metric define a singular Hermitian metric on (\mathscr N\otimes K_U^{-1})^{\otimes q_0}. Its q_0-th root is a metric on the actual line \mathscr N\otimes K_U^{-1}, with multiplier ideal \OO_U and curvature dominating a positive multiple of the pullback Fubini–Study form. The same statements hold after every nonnegative twist by f^*A_Y.

The injectivity theorem [34, Theorem A] applies on the compact Kähler U: multiplication by the pullback of any nonzero section of a positive power of A_Y is injective on

into the correspondingly higher twist. Its assumptions are exactly the semipositive smooth metric on the multiplying line, the preceding positive lower curvature bound, and the trivial multiplier ideal. This injectivity implies Equation (5.18). For the second assertion, the Leray edge map injects H^1(Y,f_*\mathscr N) into H^1(U,\mathscr N). Multiplication by a section of a sufficiently high A_Y-power sends this subspace to zero, since Serre vanishing kills H^1(Y,f_*\mathscr N\otimes A_Y^j) for large j. Injectivity upstairs forces the original subspace to be zero. For the first assertion, suppose R^1f_*\mathscr N has a nonzero torsion subsheaf. A section of a sufficiently high power of A_Y annihilates a nonzero such subsheaf. After a further sufficiently large twist it has a nonzero global section, and Serre vanishing and Leray identify that section with a nonzero class in H^1(U,\mathscr N\otimes f^*A_Y^j). Multiplication annihilates this class, again contradicting injectivity. This proves both assertions.

The resulting surjectivity extends the floor sections as follows. Pull any section of uL|_{D_Z} to C, multiply by the canonical section of E|_C, and extend by this surjectivity. The extended section descends to Z, since r_*\OO_U(E)=\OO_Z by exceptionality and normality. On the strict transforms of the components of D_Z, division by the canonical section of E recovers the prescribed section. The descended restriction therefore equals it on the whole reduced D_Z, since equality holds generically on every component. Since uL|_{D_Z} is generated, this proves Equation (5.16).

We finish by showing that this stable base locus is empty. The base loci of factorial Cartier multiples form a descending sequence of compact analytic subsets and hence stabilize. Sections exist because H_Y is big and Equation (5.15) descends their pullbacks. Choose a divisible degree v whose base locus is \mathbf B(L), and let \mathfrak b be its base ideal. If \mathbf B(L)\ne\varnothing, the ideal is a unit near D_Z, while (Z,\Delta_Z) is klt outside D_Z. Its log canonical threshold

is therefore a positive rational number, attained by a divisor whose center is contained in \mathbf B(L). This follows directly on a simultaneous log resolution of the pair and the ideal: the threshold is the minimum of the positive rational discrepancies divided by the positive integral ideal orders.

Choose an integer s>t and general divisors D_1,\ldots,D_s\in|vL|. On the same resolution their fixed part is the divisor of \mathfrak b; their free transforms are jointly transverse to the SNC data. Hence

is log canonical: the fixed part is at its log canonical threshold and the free components have coefficients t/s<1. There is a new lc place centered in \mathbf B(L), and the new adjoint satisfies the actual identity

Apply [38, Theorem 3.3] to this compact Kähler lc rational pair. It gives a projective strongly \Q-factorial dlt modification d:Z'\to Z. Since the pair is lc, all extracted divisors have log discrepancy zero in our convention, so the modification is crepant. The source is again compact Kähler. Choose its defining dlt log resolution, with exceptional crepant coefficients strictly below one and an isomorphism at the general point of every lc stratum; resolving the remaining data away from those points gives the resolution required by Definition 3.1. The floor on Z' has a point over the center of this lc place, hence over \mathbf B(L). The new nef adjoint is d^*L_{\mathrm{new}}. On a common resolution with U, it satisfies Equation (5.15) with the big nef line (1+tv)H_Y. The previously proved assertion (5.16) therefore makes its stable base locus disjoint from its floor. On the other hand, normality and projection formula identify all divisible sections under d, and give

This locus contains the stated floor point, a contradiction. Thus \mathbf B(L)=\varnothing; factorial stabilization provides an actual globally generated multiple of L.□

Proof of Proposition 5.1. By Proposition 2.7 we can begin on the resolved source in the statement. We use a secondary induction on the positive base dimension k. Apply Lemma 5.2 and then Lemma 5.3 to obtain the prepared data and the pseudo-effective line H.

If a(W)=0, Lemma 5.4 and the first case of Lemma 5.8 give J\sim_{\Q}N(J), which is already the required decomposition with trivial positive part. If W is projective, apply Lemma 5.7. Its first alternative has a strictly smaller positive base dimension, so the secondary induction and birational transfer finish the proof. In its second alternative, take a common smooth base resolution in Equation (5.8) and prepare the source over it. Equation (5.4) adds an effective divisor exceptional over the old base. It is also exceptional over W_m, because the base program extracted no divisors. Thus the second case of Lemma 5.8 applies.

We have obtained

with P_0 nef. A rationally trivial P_0 finishes the proof. In the remaining case P_0=b^*H_m, where b:X\to W_m is proper and H_m is big and nef on the projective W_m. Apply Proposition 3.8 to this nef-plus-negative presentation. It gives a globally strongly \Q-factorial dlt compact Kähler model (Z,\Delta_Z) with nef adjoint L_Z. The ordinary dlt endpoint has the resolution property of Definition 3.1. On a common smooth resolution \mu:U\to X, r:U\to Z its exact positive-line comparison is

Proposition 5.9 makes L_Z semiample. Hence \mu^*P_0 is semiample. Lemma 2.4 identifies \mu^*A=N(\mu^*J), so this is precisely \Ggood_n on U. Finally Proposition 2.7 transfers the decomposition back through all the prepared source modifications.□

Proof. If the Albanese map of X is nonconstant, simplicity makes its image n-dimensional: a positive-dimensional general fiber of smaller dimension would be a forbidden subvariety. At a point where the Albanese map has rank n, some n invariant one-forms have nonzero wedge after pullback. This gives a nonzero holomorphic section of K_X, contrary to our assumption. Thus H^1(X,\OO_X)=0. The exponential sequence embeds \Pic(X) into the countable group H^2(X,\Z).

The manifold X is not uniruled, by simplicity. Ou’s non-pseudo-effectivity criterion for the canonical class therefore gives L\ge0 [58, Theorem 1.1]. We recall how the slope and foliation results in the same paper give the more precise inequality in Equation (6.1). Let \gamma be any class in the full dual of the pseudo-effective cone, and use the slope

for torsion-free sheaves. If \mu_{\gamma,\min}(\Omega_X)<0, the first Harder–Narasimhan piece \mathscr T\subset T_X has strictly positive slope. It is saturated and semistable. The tensor slope inequality shows that the bracket \bigwedge^2\mathscr T\to T_X/\mathscr T is zero: the minimum slope of its source is at least 2\mu_\gamma(\mathscr T), whereas the maximum slope of its target is smaller than \mu_\gamma(\mathscr T). Consequently \mathscr T is a foliation. Its dual has strictly negative maximum slope and is not pseudo-effective by [58, Proposition 4.5]. Ou’s foliation theorem [58, Theorem 1.4] makes this foliation the tangent foliation of a meromorphic fibration with compact general leaf closures. Its rank is strictly between zero and n: rank n would give \mu_\gamma(T_X)=-L\cdot\gamma/n\le0. The general leaf closure is therefore a positive-dimensional proper subvariety through a general point, again contradicting simplicity.