Finite ordinary minimal model programs on compact Kähler fourfolds

Proof. The pullback assertion is [24, Lemma 2.39]; nef descent is [6, Lemma 2.38]. In particular, the first assertion applies to any proper bimeromorphic morphism between compact spaces with rational singularities whose source is in class \mathcal C: its general fibre is a point. For a projective klt log Fano contraction, relative vanishing supplies the higher-direct-image hypothesis once rationality of the base has been established.□

Proof. Choose a smooth representative \theta of \alpha with local smooth potentials. On the smooth compact Kähler space U, a positive representative of the given class has the form

for a global distribution v. Remove from T its singular locus and the image of the exceptional locus, obtaining a smooth open set T^\circ whose complement has codimension at least two. The map p is an isomorphism there. Equation (2.1) gives a global \theta-plurisubharmonic function v^\circ on T^\circ. We do not assert that v is quasi-plurisubharmonic along E.

On a coordinate neighbourhood where \theta=\ddc\rho, extend \rho+v^\circ first across the removed codimension-two subset of the regular locus, and then across the singular locus. The first extension is the plurisubharmonic Hartogs theorem; the second is its normal-space version of Grauert–Remmert, recalled in [5, Section 2.1, pp. 927–928]. These extensions are unique. On overlaps their differences are the original smooth pluriharmonic differences of the functions \rho. Subtracting \rho therefore gives a global potential v_T, and

is a current with local plurisubharmonic potentials representing exactly \alpha. This proves descent, including for signed E.

For pullback, compose local plurisubharmonic potentials with the resolution. They cannot become identically -\infty on a nonempty open subset, since a resolution is an isomorphism on a dense open set. Thus they define the pulled-back positive current; see [5, Section 2.1, p. 928].

Finally, if \alpha is nef, then p^*\alpha is nef on U. For a Kähler form \omega_U, choose positive representatives of p^*\alpha+\varepsilon[\omega_U]. Their masses are bounded as \varepsilon\downarrow0, so a weak limit is a positive current in p^*\alpha. On the smooth Kähler manifold U it has local plurisubharmonic potentials. The first part, with E=0, descends this current to the required class on T.□

Proof. Part (i) is the analytic negativity lemma [24, Lemma 2.5]; apply it to both signs for the last assertion. For (ii), points of a connected projective fibre can be joined by a chain of irreducible curves. If the fibre met both \Supp E and its complement, some curve in such a chain would meet the support without being contained in it. Its intersection with the effective real Cartier divisor would be strictly positive, contrary to f-anti-nefness. The assertion for a real Cartier divisor follows locally from positive combinations of effective rational Cartier divisors, as explained below.

For (iii), let G be the normalization of the graph in T\times_ZT^+, with projections p_G,q_G and map h:G\to Z. Compatible canonical data give the actual real Cartier divisor

This difference is defined locally by compatible meromorphic canonical generators and therefore glues, even when the canonical sheaves were specified as \Q-line bundles. Smallness makes it exceptional over both sides. It is effective by (i), applied over T: on a p_G-vertical curve the degree of -F_G is the degree of q_G^*D^+, which is nonnegative.

For every h-vertical curve C one has

Indeed, normalization is finite onto the graph, so the two projections cannot both contract C. The nonzero degrees contributed by p_G^*D and -q_G^*D^+ are both negative. An effective real Cartier divisor has nonnegative degree on a curve outside its support. For completeness, this fact and its strict pullback analogue hold even if its individual prime components are not \Q-Cartier: locally express it in the real span of finitely many Cartier divisors. Vanishing of coefficients outside its support and nonnegativity of the remaining coefficients are rational linear conditions. The resulting rational polyhedral cone expresses it as a positive real combination of effective \Q-Cartier divisors. The usual effective Cartier assertions apply to each term; summing the nonnegative local intersection numbers gives the asserted global degree inequality. In particular, the pullback has positive coefficient at every prime whose centre is contained in the support.

The morphism p_G has connected fibres because T is normal; hence h=f p_G has connected fibres. Its fibres are projective because G is finite over the graph in the fibre product. Each point of a positive-dimensional fibre lies on a curve in that fibre. Equation (2.4) therefore places the entire fibre in \Supp F_G. To identify the relevant fibres, suppose f is an isomorphism over an open subset of Z. There the log divisor on T^+ is pulled back from Z, by smallness and agreement in codimension one. Relative ampleness excludes an f^+-vertical curve there; projectivity, connectedness and normality then make f^+ an isomorphism there as well. Interchanging the sides proves that their exceptional images coincide. Thus h^{-1}(A)\subset\Supp F_G.

Pull F_G back to a resolution on which P appears. Its coefficient at P is exactly a(P,T^+,B^+)-a(P,T,B), with our log-discrepancy convention. Effectivity gives monotonicity, and the support assertion gives the claimed strictness. This proves (iii).

For (iv), a small exceptional locus in a fourfold has dimension at most two and has positive-dimensional fibres, so its image has dimension at most one. Moreover F_G\ne0 by (2.4). A component of its support has dimension three and maps finitely into \Exc(f)\times_Z\Exc(f^+). The dimension of this product is at most the sum of the dimensions of its two factors. This proves (iv).

For (v), choose a common smooth resolution p:V\to T, q:V\to T' carrying the fixed b-datum and projective over the common contraction base. Such a resolution is obtained by taking projective modifications of the graph that resolve the maps to a carrier of the datum. Subtract the actual structure boundaries to obtain a real Cartier divisor F=B_{T,V}-B_{T',V}. Its coefficients are the differences of generalized log discrepancies, and cancellation of the fixed b-part gives

Let h:V\to Z be the morphism to the common contraction base; in the divisorial case Z=T' and q=h. It is projective by this choice and has connected fibres because it is bimeromorphic onto the normal space Z. The boundary-pushforward condition makes F exceptional over Z. The relative signs of the log classes give F\cdot C\leq0 on every h-vertical curve, with strict inequality whenever p(C) or, in the small case, q(C) is a curve. Thus (i) gives F\geq0.

If y lies in an exceptional image, a vertical curve downstairs can be lifted to a curve on V dominating it: use a component of its projective inverse image and cut by relatively ample hyperplanes. Such a curve has negative F-degree and hence lies in \Supp F. Part (ii) now puts the entire fibre h^{-1}(y) in the support. Pullback to a higher resolution has positive coefficient at every prime centred in this support. This proves all the strict discrepancy inequalities, including for a centre on the flipped side. No trace current on the normalized graph is assumed to be Cartier; the actual divisor is constructed from the structure boundaries on the smooth resolution.□

Proof. On every p-vertical curve, -J_T has the same degree as the nef class q^*H. Lemma 2.3(i) gives J_T\geq0. Subtract two possible relations. The difference of the correction divisors is p-exceptional and numerically trivial over T, hence zero. Pullback injectivity from Lemma 2.1 then gives uniqueness of the class as well.

Pulling back (2.6) to a higher resolution gives the claimed correction there by uniqueness. Conversely, if the relation is initially known on a higher resolution r:V'\to V, push its correction down to V. The difference between the upstairs correction and the pullback of this pushforward is r-exceptional and numerically trivial over V, so is zero. Injectivity of r^* gives the relation on V.

Equation (2.7) follows by adding K_V and the fixed nef class tq^*H to its two sides. Its coefficients give

on every sufficiently high resolution. This proves the discrepancy assertions.

For the converse, let B_V be the given actual generalized structure boundary and put B_V^{\rm ord}=p^*(K_T+B_T)-K_V. Both push forward to B_T, so

is an actual p-exceptional divisor. Subtract the ordinary log class from (2.2) to obtain (2.6). This constructs the divisor before applying negativity; it does not infer a divisor from an arbitrary class difference.□

Proof. The positive current q^*h+[J_T] represents p^*H_T. The proof of Lemma 2.2 constructs a positive current S_T in H_T with local potentials. On the big open where p is an isomorphism and the correction is absent, it agrees with p_*q^*h. Their difference is a closed current of order zero supported in codimension at least two. Such a current vanishes: after a local embedding it has bidimension (\dim T-1,\dim T-1), whereas its support has smaller dimension [9, Chapter III, Corollary 2.11]. Thus S_T is precisely the actual trace in (2.5). This step is needed because a trace of a positive b-current need not have local potentials a priori [7, Remark 2.6(ii)].

The class q^*H is nef and has positive top self-intersection: q^*h is semipositive and is positive on the dense open where q is an isomorphism. By [10, Theorem 0.5], it contains a Kähler current R\geq\delta\omega_V for some \delta>0 and some Kähler form \omega_V. Fix a Kähler form \omega_T and choose C>0 with p^*\omega_T\leq C\omega_V. For 0<\varepsilon<\delta/C,

Its class is p^*(H_T-\varepsilon[\omega_T]). Lemma 2.2 shows that H_T-\varepsilon[\omega_T] is pseudo-effective. Hence H_T contains a Kähler current, as required.□

Proof. Subtract the two structure equations on V. The trace of the fixed b-datum cancels, giving (2.8). At a prime dominating a divisor on T', the corresponding boundary coefficient on T is unchanged, so its coefficient in F is zero. This proves exceptionality over T'. Smallness gives the same assertion over T.

If L_{T'} is nef, then -F is p-nef, because its degree on a p-vertical curve is the degree of q^*L_{T'}. Negativity gives F\geq0. If L_T is also nef, then F is q-nef, so applying negativity with the opposite sign gives F\leq0. The same proof uses only relative nefness when the models lie over a common base. For equal pullback classes, the already constructed exceptional divisor is relatively numerically trivial, and is zero by Lemma 2.3(i).□

Proof. Fix z\in Z and shrink to a connected Stein neighbourhood U on which the stated pair is defined. A line bundle M on f^{-1}U has a Cartier-divisor representative there: the coherent direct image f_*M has rank one over the nonempty open where f is an isomorphism, so Cartan’s Theorem A supplies a nonzero section. Its pullback is a section of M that is not identically zero, and its zero divisor is Cartier. This argument takes place over U, not on the whole compact space.

Every rational line bundle N on f^{-1}U is also relatively big. Indeed, choose q>0 with qN a line bundle and an f-ample line bundle A. The same direct-image argument gives a nonzero section of qN-A, hence

This is relative bigness in the analytic sense [15, Definition 2.46]. Apply it to N=L-(K_T+C_U). The klt base-point-free theorem gives, after shrinking around z, relative generation of L^k for every sufficiently large integer k [15, Theorem 6.2 and Remark 6.3].

Choose finitely many such generating sections near the compact fibre F=f^{-1}(z). The morphism they define on the projective reduced fibre F_{\mathrm{red}} is constant on each irreducible component: a positive-dimensional image would yield a curve with positive degree against the pullback of the hyperplane bundle, contrary to the numerical triviality of L^k. Such a curve is obtained by successive general hyperplane sections of a component. Connectedness of F makes the images of all its components the same point. A linear combination of the generating sections is consequently nonzero at every point of F_{\mathrm{red}}.

This section is a generator of L^k in the local ring at every point of F: its residue is nonzero, and hence its representing function is a unit. This also treats a nonreduced fibre. Properness allows us to shrink U away from the image of the section’s zero set. Thus L^k is trivial on f^{-1}U. Apply the argument to consecutive integers k and k+1, and use one common neighbourhood. Their quotient trivializes L itself. Since f_*\cO_T=\cO_Z, on this neighbourhood f_*L\simeq \cO_U, and evaluation is an isomorphism. These local conclusions give (3.1) globally. No additional tensor power enters the integral conclusion.□

Proof. Fix a Kähler class \omega and write d=c_1(D). Cone duality applies because klt singularities are rational. The slice

is compact. Indeed, fix smooth representatives of a basis of \BC(T). Each is bounded above and below by a multiple of a Kähler form, so all its pairings on S are bounded; the slice is closed. If D is not nef, the minimum of d on S is negative. An extreme point of its minimum face gives a negative extremal ray.

Fix any such ray and let r\in S be its normalized point. Choose \varepsilon>0 with (d+\varepsilon\omega)\cdot r<0. The descending nef part \varepsilon\omega changes no discrepancy, and B+\varepsilon\omega is big since B is effective and \Q-Cartier. The finite form of the cone theorem gives

where the C_j are rational curves. Normalize their classes in S and discard repetitions. By extremality, r is one of these points, say r_1. Let K be the convex hull of the others and of S\cap\{d+\varepsilon\omega\geq0\}. This is compact and excludes r; otherwise extremality of r would fail.

If K is nonempty, strict separation gives a linear functional h with h(r)=0 and h>0 on K, after subtracting a multiple of \omega. Regard it as a Bott–Chern class by duality. For all sufficiently small \delta>0, h-\delta d is strictly positive on K and at r. Since S=\operatorname{conv}(\{r\}\cup K), it is strictly positive on S, hence Kähler. Thus \alpha=h/\delta has the required properties. Its positive minimum on K persists under small perturbations in R^\perp; openness of the Kähler cone also preserves \alpha-d. These perturbations give \mathcal U_R. If K is empty, then S=\{r\} and -d is Kähler; a small neighbourhood of 0 in R^\perp has the required properties. This includes the case R^\perp=\{0\}.□

Proof. Clear the denominator of L. Its Bott–Chern identity permits a smooth Hermitian metric whose curvature is the indicated difference of forms. On every fibre its curvature is positive. Positivity of a line bundle on a compact complex space implies ampleness, and for a proper analytic map ampleness on every fibre is equivalent to relative ampleness [18, Corollary 1.12 and Definition 3.1–Remark 3.2]. This criterion does not presuppose projectivity and applies to nonreduced fibres as well. Existence of the relatively ample line bundle makes f projective.

For the converse, local relative embeddings give metrics on a relatively ample bundle with positive curvature in fibre directions. Glue their weights by a partition of unity on the base. The resulting metric retains that positivity. Add a sufficiently large multiple of a base Kähler form to its curvature. The sum is strictly positive; compactness allows one multiple to work everywhere.□

Proof. Choose \alpha\in\mathcal U_R from Lemma 3.2 and put \kappa=\alpha-c_1(D). Proposition 6.23 gives f:T\to Z and \gamma Kähler with \alpha=f^*\gamma. Thus c_1(-D)=\kappa-f^*\gamma, and Lemma 3.3 proves projectivity and relative ampleness. This establishes projectivity without strong factoriality. For every curve C\subset T,

The rational curve generating R is contracted, so f is nontrivial.

Relative vanishing gives R^j f_*\cO_T=0 for j>0 [15, Theorem 5.2]; the log Fano base has rational singularities [6, Lemma 8.8]. Lemma 2.1 now identifies f^*\BC(Z) with the classes of degree zero on all vertical curves, which is exactly R^\perp. The global line-bundle rank is also one, since all those curves are proportional and the rational line bundle D detects their ray. This proves (3.4). The lower-dimensional case has all the stated Mori fibre-space properties. In equal dimension, connectedness gives generic degree one, so f is bimeromorphic.

Divisorial contractions. Suppose E is an exceptional prime. It is \Q-Cartier. If E\cdot R\geq0, it would be f-nef, contradicting negativity for a nonzero effective exceptional divisor. Thus E\cdot R<0. Every vertical curve lies in E, since an effective \Q-Cartier divisor has nonnegative degree on a curve outside its support. Every positive-dimensional projective fibre is covered by curves. A zero-dimensional local component of a connected fibre would be isolated, and the birational form of Zariski’s main theorem would make f an isomorphism there. Consequently \Exc(f)=E.

For a prime Weil divisor A_Z on Z, let A_T be its strict transform. Choose t\in\Q so (A_T+tE)\cdot R=0. The coefficient is rational by taking the ratio of the two rational degrees on one integral curve. A Cartier multiple of A_T+tE descends by Lemma 3.1: its difference from D is relatively ample. The descended line bundle agrees with the corresponding reflexive power of \cO_Z(A_Z) off f(E), a subset of codimension at least two. Reflexivity identifies them everywhere. Hence every global prime on Z is \Q-Cartier.

If a global canonical Weil divisor is given, push it forward. It agrees with the canonical sheaf on the big open where f is an isomorphism, so reflexivity shows that it remains a canonical Weil representative. Alternatively, start with an invertible reflexive power of \omega_T, adjust it by a rational multiple of E to degree zero, and descend after clearing denominators. On the same big open the result is a reflexive power of \omega_Z; hence that power is invertible everywhere. This proves the canonical assertion without assuming a global meromorphic canonical section.

Now D' is a rational line bundle and its canonical identification away from E gives (3.5). Intersection with R gives e>0. If b_E is the coefficient of E in B, then

Pullback of the effective divisor eE gives weak increase for all other log discrepancies. Thus the target pair is klt.

The small canonical model. Suppose f is small. Finite generation of the ordinary relative log canonical ring [15, Theorem 1.18] gives local finite generation of (3.6). When canonical data are written as sheaves, this application can be made over each Stein open in Z: nonzero sections of rank-one proper direct images provide the meromorphic representatives there. Changing representatives induces compatible graded isomorphisms of the intrinsic algebra.

The ordinary analytic flip theorem [15, Theorem 1.14] constructs precisely its relative canonical model. It has no local factoriality hypothesis. Its proof glues the unique local log canonical models, giving a normal small projective f^+:T'\to Z globally. The fibres are connected because Z is normal. A finite cover of the compact base permits one common sufficiently divisible Veronese generated in degree one. Replace r by that multiple. The resulting tautological bundle is relatively ample and agrees with \cO_T(rD) on the common big open. Its reflexive extension is \cO_{T'}(rD'), proving (3.7) as an actual bundle isomorphism. This also proves that D' is a rational line bundle. The relative Proj is invariant under this Veronese operation. By Lemma 3.3, T' is compact Kähler.

For a global prime A' on T', transform it to A on T and choose t\in\Q so (A+tD)\cdot R=0. Descend a Cartier multiple by Lemma 3.1. On T' its pullback agrees on the common big open with the same multiple of A'+tD'. Reflexivity extends the identification; since D' is already a rational line bundle, A' is \Q-Cartier. This proves global Weil-divisor factoriality. A given canonical Weil divisor is transported on that common open and then reflexively. In the sheaf formulation, repeat the descent argument with an invertible power of \omega_T adjusted by tD; undoing the twist by tD' proves the rational-line-bundle property of \omega_{T'}.

On a common smooth resolution p:V\to T, q:V\to T', Lemma 2.3 gives the actual discrepancy divisor

exceptional over both sides, with the asserted strictness over the flipping and flipped loci. This also proves klt preservation.

Optional strong factoriality. In the divisorial case, take any coherent rank-one reflexive sheaf \cF on Z. Its reflexive pullback to T is coherent of rank one. Strong factoriality supplies an invertible reflexive power. Adjust it by a rational multiple of E, descend as above, and compare off f(E). A reflexive power of \cF is therefore a line bundle.

In the small case start with such a sheaf \cF' on T'. On a common smooth resolution put

Here \cM is a line bundle and \cF is coherent reflexive of rank one; both give the required transform on the common big open. An invertible power of \cF, adjusted by a rational multiple of D, descends. Pull back the descended bundle and undo the twist by D'. After clearing denominators, reflexivity identifies the resulting line bundle with a reflexive power of \cF'. Thus the strong condition is preserved. Only sheaves defined on the whole compact spaces have been used.□

Proof. Since D\cdot R\ne0, subtracting a unique multiple of c_1(D) makes \theta vanish on R. Equation (3.4) and pullback injectivity give the unique \eta. On a common resolution over Z the two pullbacks of \eta agree. Therefore

The expression in brackets is the actual exceptional discrepancy divisor already constructed in the proof of the proposition.

If \theta=c_1(A) for a global Weil divisor A, the difference p^*A-q^*A' is likewise exceptional over T', where A' is its transform or pushforward. Subtracting the two identities represents q^*(\theta'-c_1(A')) by a q-exceptional divisor. That divisor is numerically trivial over q, so negativity in both signs makes it zero. Pullback injectivity gives \theta'=c_1(A').

Finally suppose the old trace H_T of a fixed nef class on an earlier model has, on a common high resolution, the relation

If H_T=f^*\eta+s\,c_1(D), the new correction is the actual divisor

Every old exceptional divisor remains exceptional over T': neither kind of step extracts divisors. The resulting correction J_{T'} is therefore an actual q-exceptional divisor. Its negative is q-nef because p_0^*H is nef. Negativity gives J_{T'}\geq0, and uniqueness and compatibility are those of Lemma 2.4.

For the stated real-boundary extension, Proposition 6.3 supplies the ordinary step and its negative-side Bott–Chern rank. The difference p^*D-q^*D' is an actual exceptional real discrepancy divisor, as shown in Section 6.2. The decomposition, transport formula, and correction argument above therefore apply unchanged.□

Proof. The Borel–Moore localization sequence for B\subset Y contains

The outside groups vanish: the real dimension of B is at most 2k-2. Thus j_Y^* is an isomorphism. Compose restriction from X with its inverse to obtain a linear map

For an irreducible k-cycle S\subset X not contained in A, its restriction is the fundamental class of S\cap U. Closing its image in Y gives its strict transform: analyticity follows by taking the proper image of the corresponding strict transform in the given diagram. The displayed map therefore sends [S] to this transformed cycle class. It sends [S] to zero when S\subset A.

Conversely, no k-dimensional subspace of Y is contained in B. Its strict transform in X consequently maps to its fundamental class. This proves the asserted surjection of cycle spans. If S\subset A has dimension k, its nonzero class, by (4.1), is in the kernel. This proves the strict inequality.

For a transformation extracting no prime divisor, choose the common isomorphic open so that the complement in the target has codimension at least two. Every contracted source prime lies in the source complement. Apply the result with k=n-1 at every step. The nonnegative integer c_{n-1} cannot decrease infinitely often. For a small transformation both complements have codimension at least two, so applying the result in both directions gives equality.□

Proof. Let L\in\BC(X) be the generalized log class and let M_W=[\bM_W] be the nef trace class on W. Write the crepant formula as

This is the crepant boundary identity defining generalized discrepancies; it is an equality of Bott–Chern classes when the nef data are transcendental. The boundary \Delta is an actual real divisor with simple normal crossing support. Since the nef data descend to W, every further generalized discrepancy is the ordinary discrepancy for (W,\Delta). The coefficients of \Delta need not be nonnegative. We may enlarge its listed simple normal crossing divisor by components of coefficient zero. Here a closed stratum means an irreducible component of an intersection of listed components; it is smooth, and is the closure of its open stratum.

We give the local estimate and the realization argument explicitly; they are the log smooth calculations underlying [31, Proposition 2.36] and [3, Proposition 2.8]. Write \Delta=\sum_j d_jD_j and put w_j=1-d_j. Suppose first that every w_j is positive, and set

Let E be exceptional over a smooth model with this log smooth boundary. At a general point of its center C, let c=\codim C, and choose local parameters x_1,\ldots,x_c for C so that the boundary components containing C are x_1=0,\ldots,x_r=0. For v=\operatorname{ord}_E, the Jacobian calculation gives

Indeed, on a smooth model containing E, with local parameter t for E, write x_j=t^{v(x_j)}u_j at its general point. In a top exterior differential at most one factor can use the term involving dt that lowers the order by one. Thus the relative Jacobian has order at least \sum_{j=1}^c v(x_j)-1. Adding one and subtracting the boundary orders proves (4.3). The remaining coordinates along C are holomorphic and contribute no negative orders.

In particular every discrepancy is at least \delta: for a divisor already on the smooth model it is its listed weight, or one if it is not listed; for an exceptional divisor use (4.3). If C is not a stratum of the listed simple normal crossing divisor, then c>r. When r>0, the right hand side is at least \delta+1; when r=0, it is at least c\geq 2\geq\delta+1. Hence

Now fix a place E with discrepancy strictly below 1+\delta. If it is not already a divisor on W, its center is a stratum by (4.4). Blow up that closed stratum. The center is smooth, the blowup is projective, and the new listed boundary is again simple normal crossing. If the center is the intersection of the components indexed by J, then the new exceptional component has weight

Thus all weights remain at least \delta, so the same argument applies on the next model. It excludes a non-stratum center at every stage until E becomes a divisor.

This process is finite. While E is exceptional, a blowup of its smooth center of codimension c\geq2 changes its ordinary empty-boundary discrepancy by

The subtracted quantity is a positive integer, whereas an ordinary log discrepancy over a smooth space is a positive integer. Consequently only finitely many such blowups are possible. Thus E is obtained by successive stratum blowups, which is the meaning of a toroidal place here. This also realizes it on a projective model without first assuming that the original model on which E was given was projective over X.

For completeness, there are only finitely many such toroidal places. A toroidal prime over a fixed irreducible stratum is determined by a primitive nonzero vector (m_j)_{j\in J} of nonnegative integers in the normal boundary directions. Its discrepancy is

These statements follow also directly from the successive stratum blowups: a blowup inserts the sum of the relevant coordinate rays, and the discrepancy transforms by (4.5). The vector determines the monomial order and is unchanged when a boundary equation is multiplied by a unit. There are finitely many irreducible strata on the compact model. Since each w_j\geq\delta, an upper bound T on (4.6) bounds every m_j by T/\delta. This leaves finitely many vectors and hence finitely many places. Add the finitely many prime divisors already on W which are exceptional over X.

For a generalized klt pair, all weights in (4.2) are positive. The preceding argument with \epsilon=\delta proves (i) and its projective realization assertion. In particular a\leq1 lies strictly below 1+\delta.

No rationality of the weights was used: their fixed positive minimum bounds the nonnegative integer vectors. Thus the conclusion applies to real boundaries and real nef data.

For a generalized terminal pair, the coefficient of every p-exceptional component in \Delta is negative, since its log discrepancy is greater than one. The other coefficients are the original boundary coefficients and are at most b<1. All the weights on W are therefore at least 1-b. Repeat the argument with \delta=1-b and T=2-b to prove (ii). Subsequent stratum blowups preserve this lower bound by (4.5).

For (iii), the weights on W are nonnegative. Set

The coefficient-one components have image in X\setminus U. Thus, at the general center on W of a place whose center on X meets U, no zero-weight component occurs. Work there over U. The same estimates and stratum blowups use only positive weights, so give the uniform lower bound and the strict cutoff 1+\delta. The strata in question are restrictions of the finitely many global strata, and the blowups can be performed along their smooth closed closures on the compact model. Meeting a zero-weight component elsewhere does not change a weight computed at the general point of the center. This proves finiteness and projective realization also in this case.

Finally, to represent the entire finite set at once, take the dominating component of the fiber product of the finitely many projective realizations over W, normalize, and resolve projectively. The strict transforms of the designated prime divisors remain divisors on this common model. Its composite map to X is projective.□

Proof. Dropping the effective boundary shows that T has terminal singularities. Suppose its singular locus has a codimension-two component. Near a general smooth point of this component choose a holomorphic projection to \C^2 submersive along it. Take a general fibre transverse both to the regular locus and, on a fixed log resolution, to the exceptional divisor and its strata. These transversality conditions hold after shrinking and choosing a general value. The resulting surface is a complete intersection in the Cohen–Macaulay space T, since klt singularities are rational, and is regular outside isolated points. It is therefore normal.

Adjunction of a local Cartier pluricanonical power, first on the regular locus and then by reflexivity, restricts the relative canonical equality to the surface resolution. All its exceptional discrepancy coefficients are positive. A normal surface germ with this property is smooth. Indeed, negative definiteness of the exceptional intersection matrix shows that a positive exceptional discrepancy divisor has negative intersection with some exceptional curve. That curve has negative canonical degree; adjunction makes it a smooth rational (-1)-curve. Contract it smoothly and repeat. The remaining discrepancy coefficients stay positive by pushforward. Eventually no exceptional curve remains, and the proper bimeromorphic map to the normal surface is an isomorphism.

On the other hand, a slice of a singular fourfold point by two equations has embedding dimension at least the ambient embedding dimension minus two, and hence greater than two. The chosen surface is singular there, a contradiction.□

Proof. Choose a log resolution on which the positive strict boundary components are smooth and mutually disjoint. Starting with simple normal crossing support, blow up intersections of the positive components to separate them. Every exceptional crepant coefficient is negative by terminality. Thus the only positive components on this resolution are those disjoint strict transforms, with their original coefficients.

Consider a place not already on the resolution. At its general centre, apply (4.3), dropping negative boundary coefficients. A centre of codimension at least three has log discrepancy at least 3-b>2, if it lies in a positive component of coefficient b, and at least three otherwise. A codimension-two centre outside the positive support has log discrepancy at least two. Thus a place with a<2 has codimension-two centre in one positive component, and its log discrepancy is at least 2-b.

If this centre lies in the exceptional locus, it is an irreducible component of the intersection of that positive component and an exceptional divisor. There are finitely many such centres on the compact resolution. We check that each supports only finitely many places with a<2. It suffices to retain only the smooth component of coefficient b, since deleting the negative coefficients lowers discrepancies. Blow up the fixed centre at its general point. Unless the given place has appeared, the same estimate forces its next centre to be the intersection of the positive strict transform and the new exceptional divisor. This intersection is unique over the general point of the original centre. The successive exceptional log discrepancies are

At each further stage the negative coefficient of the newest exceptional divisor adds k(1-b) to the lower bound 2-b. Hence only k<1/(1-b) can occur before the cutoff two is reached. This proves finiteness over every fixed exceptional centre. These generic blowups are realized globally by blowing up the closures of the centres.

Add the finitely many exceptional primes on the chosen resolution. All other centres under consideration lie outside its exceptional locus at their general points, where the resolution is an isomorphism. Their images are precisely the surfaces described in the statement.□

Proof. By Lemma 2.3, discrepancies do not decrease, and they increase strictly for a place centred in either exceptional locus. Since the maps are small, the exceptional places are the same on all models. Thus every pair remains terminal in the stated sense, and its ambient fourfold is smooth in codimension two by Lemma 4.4.

Write the distinct positive coefficients of the boundary in decreasing order. We successively remove, by discarding finitely many initial steps, every flipping and flipped surface contained in a component of each coefficient. After the positive coefficients we treat the value zero, at which stage every flipped surface is considered.

Eliminating flipped surfaces. Fix the current coefficient b, and suppose that no flipping or flipped surface is contained in a component of coefficient greater than b. For b=0 assume that all positive coefficients have already been treated. If S is a flipped surface contained in a coefficient-b component, or any flipped surface when b=0, blow it up at its general point. The exceptional prime E has new log discrepancy

where \Phi_{l,i+1} are the positive prime boundary components and b_l their fixed coefficients. The blowup place is defined globally by blowing up the closed surface and resolving; its generic point suffices for this computation.

The values in (4.7) belong to a finite set, even for irrational coefficients. Indeed, terminality gives

Each multiplicity is an integer, and there are finitely many positive components. We use no discreteness assertion for all discrepancies.

Strictness in Lemma 2.3 gives a(E;T_i,\Phi_i)<2-b. Its discrepancy on the first model of the current tail is therefore also strictly below 2-b. By Lemma 4.5, outside a finite set such a place would initially have a surface centre in one boundary component of coefficient c, with discrepancy at least 2-c. Necessarily c>b. That centre persists as a surface in this component at every later step: at each step the map is an isomorphism at its general point, since otherwise the centre itself would be a flipping surface in a higher-coefficient component. It cannot become a flipped surface either, by the same higher-coefficient exclusion. Thus it cannot be the place just constructed. All these witness places belong to a fixed finite set.

Each member of that finite set is used only finitely often. Between uses its discrepancy does not decrease, and at each use it strictly increases to one of the finitely many values (4.7). Consequently only finitely many steps have a flipped surface of the current kind. Pass beyond them.

Eliminating flipping surfaces inside the boundary. Suppose b>0. Fix a coefficient-b component and normalize its transforms on the two sides of a flip. Denote the normalizations by D_i^\nu,D_{i+1}^\nu, and normalize their common image in Z_i to obtain B_i. The induced maps to B_i are proper and bimeromorphic. They are isomorphisms away from the inverse image of the common exceptional image in Z_i, whose dimension is at most one by Lemma 2.3. Finite normalization preserves this bound. On D_{i+1}^\nu its inverse image also has dimension at most one, because no flipped surface is contained in the boundary component.

Every compact surface in B_i has a strict transform on D_i^\nu, so pushforward gives a surjection

On the other side, Borel–Moore localization along the exceptional subset of dimension at most one makes restriction in degree four injective. Compatibility with proper pushforward and the common isomorphic open therefore gives an injection

These are assertions about surface spans; no surjectivity on the full fourth homology of the negative side is needed.

If a flipping surface lies in this boundary component, a surface above it on D_i^\nu maps to a set of dimension at most one in B_i. Its class is nonzero: its image in T_i is a surface, and the square of a Kähler class on T_i has strictly positive integral over it. One may compute this integral on a resolution of the surface. The finite normalization has positive generic degree on this surface, so the same nonvanishing holds upstairs. Its class is a nonzero kernel vector in the displayed surjection. Hence

with strict inequality whenever the component contains a flipping surface. Summing over the finitely many coefficient-b components shows that only finitely many such steps occur. This completes the induction at b.

The remaining flips. After the positive coefficients have been treated, perform the flipped-surface argument with b=0. No flipped surface remains. For every remaining flip, Lemma 2.3 gives

Smallness bounds both dimensions by two, so \Exc(f_i) contains a surface and \dim\Exc(f_i^+)\leq1. Apply Lemma 4.1 with k=2 to the common isomorphic open. It gives c_2(T_i)>c_2(T_{i+1}) at every remaining step, impossible for an infinite sequence of nonnegative integers.□

Proof. For flips, terminality persists by Lemma 2.3. For a divisorial contraction, (3.5) gives K_{X_i}=f^*K_{X_{i+1}}+eE with e>0. The lost prime has target log discrepancy 1+e>1; every other exceptional place has target discrepancy at least its source discrepancy, which was greater than one. Thus terminality persists throughout the sequence. Lemma 4.1 with k=3 bounds the number of divisorial steps. Any infinite sequence would therefore have a tail consisting only of flips, contrary to Proposition 4.6 with zero boundary.□

Proof. On X_0, the nef datum descends, so its addition changes no discrepancies. Thus all the initial generalized pairs are gklt. Suppose the assertions have been established through X_i. If D_i is nef, stop. Otherwise define

The set is nonempty by induction and is closed and convex. Its minimum is positive because D_i is not nef.

We first find a ray at this threshold. Choose t_k\uparrow\lambda_i with \lambda_i/2<t_k<\lambda_i. The generalized cone theorem gives an extremal ray negative for L_i(t_k). Since L_i(\lambda_i) is nef, that ray is also negative for L_i(\lambda_i/2). The trace H_i is big by Corollary 2.5. Hence \Delta_i+(\lambda_i/2)H_i is big, and the cone theorem gives only finitely many negative rays for this latter generalized pair [24, Theorem 1.3]. One ray R_i therefore occurs for infinitely many k. On this fixed ray, continuity gives L_i(\lambda_i)\cdot R_i=0. Moreover H_i\cdot R_i>0, so D_i\cdot R_i<0.

Apply Proposition 3.4 to this ordinary negative ray, and denote its contraction by f_i:X_i\to Z_i. A fibre contraction gives the stopping alternative. Otherwise make the ordinary divisorial step or flip. Lemma 3.5 transports the trace H_i to H_{i+1} while preserving its relation to the fixed b-datum. By Lemma 2.1, the class L_i(\lambda_i) descends to Z_i. Its descended class is nef, and its pullback on the new model is L_{i+1}(\lambda_i). Thus the two threshold classes have equal pullbacks on a common resolution. Their difference is represented by the actual exceptional discrepancy divisor, which vanishes by negativity as in Lemma 2.6. The step is consequently generalized crepant at the threshold.

For completeness, this also proves the induction on singularities. On a common resolution, the log discrepancy at each prime E is affine in t. Its difference across the step is zero at t=\lambda_i and is the ordinary discrepancy increase at zero. Consequently

for 0\leq t\leq\lambda_i. The generalized pairs on this interval remain gklt, and the new upper-threshold class is nef. The ordinary steps preserve all the stated properties of the original category by Proposition 3.4.□

Proof. Fix a Kähler class \omega on T and normalize \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. 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. Canonical divisors and a Cartier representative D_L of L can be chosen over U: the relevant proper direct images have rank one on the birational isomorphism locus, and Cartan’s Theorem A supplies sections nonzero there. Choose compatible canonical representatives and set

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

It is also \rho-big. Indeed, write N as a positive convex combination of finitely many rational Cartier divisors N_j in its finite Cartier span. If P is \rho-ample and q_jN_j is integral, Cartan’s Theorem A applied to the rank-one sheaf \rho_*(q_jN_j-P) gives q_jN_j\sim P+E_j with E_j\geq0. Hence N\sim_{\R}H+E with H relatively ample and E\geq0. 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 [6, 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. Pushing down the finite principal-divisor expression and pulling it back again give

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 (6.1) 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 (6.2) is f_U-antiample. Its ordinary flip and local finite generation follow from [15, 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 [15, Lemma 2.26]. The graded pieces are coherent, and the section algebra over a connected open set is an integral domain, as is seen using a meromorphic generator 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 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 (6.2) satisfy J-(K_{T_U}+\Delta_q) relatively nef. Integral descent (Lemma 3.1) 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 relative vanishing. They make T locally klt, hence rational. Leray for a projective resolution then shows that Z is rational. By [6, Lemma 8.7], f^*\BC(Z)=R^\perp and pullback is injective. 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 (6.2) 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\BC(T) 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 global divisor class this is its strict transform: the difference between the two candidate corrections is exceptional and numerically trivial over T^+, hence zero by negativity.□

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, a compact Kähler base, and

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 nef. Integral descent, Lemma 3.1, 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 (6.3) 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 (6.3) 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 [19, 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. Fibrewise ampleness is open for a proper analytic map; in the finite-dimensional span of the finitely many line bundles occurring in L, compactness therefore gives a neighbourhood of L whose rational points are relatively ample. This proves relative ampleness of L and supplies a global relatively ample rational line bundle. Clearing its denominator proves projectivity.□

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. Fix y\in Y and shrink to a Stein neighbourhood on which the boundary is defined. A Cartier-divisor representative of L exists there: h_*L is coherent and has rank one on the bimeromorphic isomorphism locus, so Cartan’s theorem A supplies a section nonzero on that locus. Its pullback is a nonzero meromorphic frame of L and gives the required Cartier divisor.

The line bundle L is h-nef. Numerical invariance of ampleness on the projective fibres gives

The analytic base-point-free theorem [15, Theorem 6.2] applies to this Cartier divisor and the real klt boundary. It states that L^m is relatively generated for every sufficiently large integer m near the compact fibre h^{-1}(y), rather than only for divisible m.

Choose finitely many generating sections. The resulting map from the reduced projective fibre to projective space is constant on each irreducible component, since a positive-dimensional image would give a curve of positive L^m-degree. The fibre is connected, so these constant values agree. A suitable linear combination of the sections therefore has no zero on the reduced fibre. In the local ring at a point of the possibly nonreduced fibre it is a unit, so it is a generator there as well. Properness allows the base neighbourhood to be shrunk so that this section trivializes L^m throughout its inverse image.

Apply this to two consecutive sufficiently large integers m,m+1. Taking the quotient of the two trivializations trivializes L. Since h_*\mathcal O_T=\mathcal O_Y, it follows locally that h_*L\simeq\mathcal O_Y and evaluation is an isomorphism. These are intrinsic assertions and hence glue over Y. This also explains why possible torsion in a fibre Picard group introduces no new index. The conclusion is recorded directly for extremal contractions in [15, Theorem 7.2(2)(iii)]; its proof allows real boundaries. The argument above gives the intrinsic global form needed here and does not require the contraction to have relative Picard rank one in every local analytic neighbourhood.□

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. The projective relative cone theorem 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. This verifies exactly the hypothesis of Lemma 6.7. 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. We first explain the one-way NQC transport that will be used. Let f_i:X_i\to Z_i be a projective ordinary K_{X_i}-negative ray contraction, and suppose that \alpha_i vanishes on its ray. The ordinary contraction theorem gives

This is Bott–Chern descent on the already projective log Fano contraction, by Lemma 2.1; the target has rational singularities and the required higher direct images vanish. We will also use injectivity of f_i^*:H^2(Z_i,\mathbb Q)\to H^2(X_i,\mathbb Q). For clarity, it follows from relative vanishing and the usual pluriharmonic-function sequence. Indeed, (f_i)_*\mathcal O_{X_i}=\mathcal O_{Z_i}, R^1(f_i)_*\mathcal O_{X_i}=0, and (f_i)_*\mathcal H_{X_i}=\mathcal H_{Z_i} imply R^1(f_i)_*\mathbb R=0; the Leray sequence gives injectivity on H^2(-,\mathbb R), and hence on rational cohomology.

Choose a weak NQC expression

for every compact curve C\subset X_i, with the \xi_{ij} in the minimal rational subspace containing \alpha_i. Since the image of f_i^* on rational cohomology is a rational subspace and its realification contains \alpha_i, that minimal subspace lies in the image. Thus

The last equality follows from injectivity. In the birational case, write g_i:X_{i+1}\to Z_i for the positive-side map, using the identity map for a divisorial contraction, and set

On a common projective resolution p:W\to X_i, q:W\to X_{i+1}, we have p^*\xi_{ij}=q^*\xi_{ij}^+. For a curve C^+\subset X_{i+1}, projectivity of q supplies a curve \widetilde C\subset W mapping onto it with some degree e>0. Consequently

Thus \alpha_{i+1} is weakly NQC. This uses a positive-degree multisection, not a degree-one lift. Its rational components may acquire different integral denominators on X_{i+1}. Nefness follows from equality of the pullbacks and descent of nef Bott–Chern classes under a proper surjective morphism. In the fibre-type case, lifting curves from Z_i through the projective morphism f_i proves weak NQC for \beta_i; its nefness again follows by descent from f_i^*\beta_i=\alpha_i.

We now choose each step separately. At a current model X_i, take any weak NQC expression as above, and choose integers l_{ij}>0 such that l_{ij}\xi_{ij} is integral. Choose

If \alpha_i=0, take any t_i>0 and omit the bound. As long as Q_i:=\alpha_i-c_1(K_{X_i}) is big and \alpha_i is not big, the class c_1(K_{X_i})+t_i\alpha_i cannot be pseudoeffective. Otherwise

would be big. The nef datum t_i\alpha_i descends to X_i, so adding it to the ordinary klt pair (X_i,0) changes no discrepancy. In particular this generalized klt adjoint is not nef, so the analytic cone theorem provides a negative analytic extremal ray R_i. Nefness of \alpha_i makes R_i an ordinary K_{X_i}-negative ray. Choose its ordinary length-bounded rational curve C_i, so that

If \alpha_i\cdot C_i>0, some \xi_{ij}\cdot C_i is positive and hence is at least 1/l_{ij}. Then

a contradiction. Therefore \alpha_i\cdot R_i=0.

Apply the ordinary rational step theorem to (X_i,0) and R_i. Its supporting nef class has the form c_1(K_{X_i})+\kappa_i with \kappa_i Kähler. Hence \mathsf B_d supplies its contraction, and the ordinary relative-positivity and flip arguments give the projective contraction and its ordinary flip when needed. All spaces remain compact Kähler, globally strongly \mathbb Q-factorial, and klt. The preceding NQC transport applies.

It remains to check the positivity hypotheses at the next stage. For a birational step, on a common resolution write

Here F_i is the actual ordinary discrepancy divisor. Thus

is big, so Q_{i+1} is big. Equality of the \alpha-pullbacks also preserves non-bigness. We can therefore recompute the NQC denominators and choose a fresh t_{i+1} by (6.5). In the non-klt application, Q_i=B_i+\mathbf M_{X_i}; the equality of the \alpha-pullbacks preserves the original generalized adjoint and its b-datum.

If this procedure never reached a fibre-type contraction, it would give an infinite sequence of ordinary rational klt K-negative birational steps. Divisorial steps are finite by the strict decrease of the span of analytic divisor classes, while small steps preserve that span, as in the cycle-rank argument of Lemma 4.1. The remaining infinite flip tail would contradict ordinary rational termination (Theorem 10.6). Hence it reaches an ordinary Mori fibre space after finitely many steps. The final \alpha-class descends to a nef weakly NQC class on its base by the fibre-type argument above. No uniform bound for the l_{ij}, and no application of a section theorem for rationally connected fibres, is required.□

Proof. We first record precisely the elementary positivity input used in passing to a smooth model. The modified-bigness resolution lemma [22, Lemma 2.5] gives a projective smooth carrier \pi:Y\to X and an effective \pi-exceptional divisor P such that, on writing

the class B_Y^++M_Y+\varepsilon P is big for every \varepsilon>0. Here M_Y is nef and B_Y=B_Y^+-B_Y^-, with disjoint effective positive and negative parts. We can replace P by an effective divisor whose support contains every \pi-exceptional prime. This positivity lemma follows directly from the definition of modified bigness, the support theorem for a modification, and the convexity of the big cone; it uses no contraction or termination theorem. All the modifications here can be chosen projective.

For clarity, if X is globally strongly \mathbb Q-factorial, that input also has the following direct verification. The class C=B+\mathbf M_X=A-K_X is a big Bott–Chern class. Put Q=B_Y^++M_Y, which is pseudoeffective. The difference

is an actual \pi-exceptional real divisor class. Given a positive divisor P supported on all exceptional primes, choose s>0 small enough that \varepsilon P-sJ\geq0. Then

is big. This uses only that pullback preserves bigness and that adding a pseudoeffective class to a big class preserves bigness.

We choose \pi to resolve the supports just used. Choose \varepsilon>0 small enough that \Delta=B_Y^++\varepsilon P has all coefficients below one. Its support is simple normal crossing, so (Y,\Delta) is klt and

The support of F_Y contains every \pi-exceptional prime, and \Delta+M_Y is big.

Smooth reduction with a Kähler nef part. Regularize a Kähler current in \Delta+M_Y and resolve its analytic singularities. This gives a projective modification q:U\to Y, with U smooth and compact Kähler, and

where \omega is a Kähler class. To obtain a Kähler class rather than a semipositive pullback, subtract a sufficiently small effective exceptional class from the smooth part and add it to G. This is the usual Kähler-current proof of the analytic Kodaira decomposition. We resolve the supports involved at the same time. Write K_U=q^*K_Y+K_{U/Y}; the divisor K_{U/Y} is effective and q-exceptional. For sufficiently small \delta>0, set

The class \beta is Kähler. At \delta=0 the second expression is the crepant boundary of the klt pair (Y,\Delta). Since its support is fixed and finite, for small \delta all coefficients of \Gamma_\delta remain below one. Its negative components are q-exceptional. If P_q is supported positively on every q-exceptional prime, choose \eta>0 small enough that

still has all coefficients below one. Its support is simple normal crossing, so (U,\Gamma) is klt. Put \mu=\pi q. We obtain

The support of E_\mu contains every \mu-exceptional prime. In particular \widetilde A is pseudoeffective.

Choose a rational effective boundary B_0 sufficiently close to \Gamma, on the same positive support, and put

Openness of the Kähler cone makes H Kähler, and (U,B_0) remains klt. Fix a Kähler form representing H and use its pullbacks as one fixed nef b-datum. Thus

where D is an ordinary rational klt adjoint. This auxiliary nef b-datum need not equal the original \mathbf M.

Scaling only above one. Choose T>1 such that D+TH is nef. Run the ordinary D-program with scaling of this fixed H, stopping as soon as the forward trace of D+H is nef. Write its traces as D_i,H_i and its successive thresholds as \lambda_i, with \lambda_{-1}=T. The construction of Proposition 5.1 applies in dimension d: at a positive threshold it uses the generalized cone theorem, finiteness of the negative rays for the big boundary-plus-nef part, and the ordinary rational step proposition. The latter uses precisely B_d together with the integral descent and relative-positivity arguments of Section 3.

More explicitly, if D_i+H_i is not nef, convexity of the nef cone gives \lambda_i>1, and there is an analytic extremal ray R_i such that

For the existence of a ray at the threshold, take parameters t<\lambda_i increasing to \lambda_i. On a ray negative for D_i+tH_i, nefness of the preceding upper-threshold class implies H_i\cdot R>0. Thus, once t>\lambda_i/2, that ray is also negative for D_i+(\lambda_i/2)H_i. The latter has only finitely many negative rays. Its boundary-plus-nef trace is big, and the pair is generalized klt by the interpolation argument below. Every step performed is therefore negative for D_i+H_i.

Each birational step is an ordinary rational klt step contracting a single analytic ray. It is projective on both sides, stays in the compact Kähler globally strongly \mathbb Q-factorial category, and has the forward class transport of Section 6.2. If F_i\geq0 is its ordinary discrepancy divisor on a common resolution, crepancy at the scaling threshold gives the discrepancy divisor

for D_i+H_i. More generally the divisor at parameter 0\leq t\leq\lambda_i is (1-t/\lambda_i)F_i. This proves the required generalized klt induction and makes the next upper-threshold class nef. It also shows that q_i^*H_{i+1}=p_i^*H_i+F_i/\lambda_i on the common resolution; in particular the forward traces H_i stay big.

A Mori fibre step cannot occur while \lambda_i>1. Indeed the pseudoeffective class D_i+H_i would be negative on every curve in a positive-dimensional fibre. Restrict a positive current in this class to a general projective fibre, and then to a smooth resolution of that fibre. Such a restriction is defined for a general fibre by the local-potential definition and Fubini’s theorem. Its intersection with a power of an ample class is nonnegative, whereas a general complete-intersection curve has class in R_i and gives a negative intersection. This is a contradiction. Pseudoeffectivity throughout the program follows either by pushing forward positive currents, or from the effective exceptional comparison and Lemma 2.2.

The program cannot have infinitely many birational steps, by ordinary rational termination in dimension at most three. If D_i becomes nef or a threshold reaches [0,1], the convex interval between it and the preceding nef upper-threshold class contains D_i+H_i. Consequently the stopping rule produces a model X^{\rm m} with

Every step actually performed had \lambda_i>1.

Removing all the divisors introduced by the resolution. Take a common projective resolution with maps r:W\to U, v:W\to X^{\rm m} and s=\mu r:W\to X. The comparisons (6.9) accumulate to a Bott–Chern comparison with an actual effective divisor

Its coefficient is positive on the strict transform of every prime on U contracted by the program. Set

By (6.7), [G_0]=v^*A_{\rm m}-s^*A. Hence G_0 is s-nef and s_*G_0=-s_*E\leq0. The negativity lemma, applied to -G_0, gives G_0\leq0. Thus

Since E is v-exceptional and E_\mu has positive coefficient on every \mu-exceptional prime, every such prime was contracted. The induced map \phi:X\dashrightarrow X^{\rm m} therefore extracts no divisor. The error in (6.10) is v-exceptional and has positive coefficient on each \phi-exceptional prime of X: the subtracted term has coefficient zero on those primes.

We now return to the original nef b-datum \mathbf M. Keep it on the same common carrier W. The generalized boundary defining the original pair on W is changed by subtracting the actual effective divisor E-r^*E_\mu in (6.10). It pushes forward to \phi_*B because \phi extracts no divisor. It defines K_{X^{\rm m}}+\phi_*B+\mathbf M_{X^{\rm m}}=A_{\rm m}, and its coefficients are below one because the original pair is gklt. Thus X^{\rm m} is a minimal model of the original generalized pair, with its original b-datum. No assertion that the original boundary remains effective or generalized klt on the intermediate auxiliary models is required.

Finally the auxiliary generalized pair (X^{\rm m},B_{0,\rm m}+\overline H) is gklt, and its boundary-plus-nef trace is big. Apply B_d to this pair to see that A_{\rm m} is semiample, so the model is good. For any original class \theta\in H^{1,1}_{\rm BC}(X), start with \mu^*\theta and use the forward transport across each ordinary analytic-ray step. This defines a class \theta_{\rm m} on X^{\rm m} and an actual exceptional-divisor comparison on W. In particular (6.10) identifies the forward trace of A with A_{\rm m}. Only this forward transport is asserted; no surjectivity from all Bott–Chern classes on the final model is used.□

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 [7, 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 regularization to choose currents with analytic singularities, arbitrarily small negative part, and zero generic divisorial Lelong number; then 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 6.12 gives N_\sigma(D_V)=F_V before this modification. Because D_V=c\alpha_V+[F_V] with \alpha_V nef, Lemma 6.15 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 6.13 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 6.14 therefore give N_\sigma(D_S)=F_S, and Lemma 6.15 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. On a projective log resolution p:W\to X carrying the nef datum, write

Strong \Q-factoriality makes K_X+B real Cartier. Thus

with J an actual exceptional real divisor. The divisor -J is p-nef, so negativity gives J\geq0. Exceptional descent of positive currents proves that \mathbf M_X is pseudo-effective. If \mathbf M_W is big, subtract a small pullback of a Kähler class on X first; the same argument proves bigness of the trace. This comparison also shows that a generalized klt pair on a strongly \Q-factorial space has ordinary klt underlying space.

Suppose now that Q is big. Regularization of a Kähler current with analytic singularities, followed by a projective log resolution, gives

Choose p also to carry the original nef datum and to resolve all divisors involved. The Kähler correction on a further blowup is obtained by subtracting a small effective exceptional divisor from the pulled-back positive part and adding it to G. Put B_W^*=G-K_{W/X}. Its pushforward is effective, and

For sufficiently small t>0, the structure data

have Kähler nef part and effective trace boundary. There are only finitely many coefficients to check; choose t so that \lfloor B_W^{\rm new}\rfloor\leq\lfloor B_W\rfloor. The resolution formula \mathcal J=p_*\cO_W(-\lfloor B_W\rfloor) proves the ideal inclusion. If the input is gklt, choose t also so that every new coefficient is less than one. On higher carriers these data are always taken by log pullback.□

Proof. For ascent, compose the contraction of M with r. A Moishezon witness is obtained by base change of the witness over M; both required maps are projective. A curve has zero pullback degree precisely when it is vertical over M or its image is a zero-degree curve on M.

For descent, the EMC map of L is constant on each r-fibre. Indeed the reduced connected projective fibre is connected by chains of curves, and all those curves have degree zero. The map therefore factors through M as a map of underlying spaces. It factors holomorphically, including on structure sheaves, because r_*\cO_L=\cO_M: locally embed the target in a polydisc and descend its coordinate functions using this equality. The factorization has the contraction property. A projective witness upstairs remains a projective witness over M by composition. Every curve of M is dominated by a curve of L: base-change to the normalization of the curve and use a multisection of the resulting projective family. The projection formula then gives the required characterization of contracted curves on M.

For a closed subspace, restrict the EMC map and take its Stein factorization. Its finite second map changes neither the contracted curves nor their degree test. Restrict the projective witness too; its map to the Stein target is projective by fibrewise relative ampleness. This proves the assertion for the full closed-subspace structure.□

Proof. The negative part P is p-exceptional, and \mathcal J=p_*\cO_W(P-F). Relative Kawamata–Viehweg vanishing gives

Indeed the difference between the actual line bundle P-F and K_W+\{B_W\} has class \mathbf M_W-p^*A, which is nef over T and big over T since p is birational. This is the usual projective-relative vanishing statement, applied locally over the base; the fractional-part boundary is ordinary klt. Because p_*\cO_W(P)=\cO_T, the divisor exact sequence yields

The last map factors through p_*\cO_F. Multiplication by the canonical section of P gives an injection \cO_F\hookrightarrow\cO_F(P): in the SNC coordinates no component of P is a component of F, even with their respective multiplicities. Surjectivity in the displayed sequence therefore identifies p_*\cO_F with \cO_T/\mathcal J=\cO_N as an algebra, not only as a module. Its support is p(\Supp F), so the induced map F\to N is surjective and is a projective contraction.

For a crepant comparison take a common projective log resolution. The actual structure subboundary there is the same on both sides, and the classes whose curves are tested have equal pullbacks. Apply Lemma 6.18 to the two contractions from the same divisor scheme F. This also applies one step at a time across flips. If F is empty, the assertion is vacuous.□

Proof. First establish the singularities of the base, independently of the pair defining \alpha. Fix a Kähler class \omega_Z on Z. Relative ampleness gives a sufficiently large t>0 for which

is Kähler. With zero boundary and this descending nef datum, X is generalized klt and its adjoint is t f^*\omega_Z. The projective canonical bundle formula [23, Theorem 2.3] makes Z generalized klt. Take its projective small strong factorialization. A small proper bimeromorphic morphism of normal surfaces is an isomorphism, so Z is globally strongly \Q-factorial. The comparison in Lemma 6.17 then makes Z klt. In particular K_Z is \Q-Cartier.

Apply Lemma 6.17 to the pair defining \alpha. This keeps the adjoint class and an effective boundary trace, and does not enlarge the non-klt locus. On a projective log resolution r:W_0\to X carrying the new data, write

The pair is generalized klt over a nonempty open subset of Z. Choose \varepsilon>0 small enough that M_{W_0}-\varepsilon(fr)^*\omega_Z remains Kähler. This changes the adjoint to f^*(\gamma-\varepsilon\omega_Z) without changing any boundary coefficient.

Choose a projective resolution \tau:S\to Z, and resolve the main component of the corresponding base change together with W_0. After a further log resolution there are projective morphisms

with W,S smooth compact Kähler, B_W having simple normal crossings support, and

Here B_W is the log-pullback boundary of the replaced pair. The map l has connected fibres: its Stein factor is finite and bimeromorphic over the normal surface S, hence is S.

Put E=B_W^-, the effective negative part of the divisor B_W. Since the boundary trace on X is effective, E is \mu-exceptional. Each of its components has image of dimension at most one in X, and consequently cannot dominate S. Choose an effective rational divisor E'\geq E with the same support, and set

The pair (W,\widehat B+M'_W) is generalized klt over an open subset of S: the horizontal coefficients are unchanged and the added divisor is vertical. Moreover \mathcal O_W(E') is a \Q-line bundle with a nonzero section in every sufficiently divisible positive power on a general fibre of l; on such a fibre it is trivial. Thus [23, Theorem 2.2], applied to

shows that

We remove the resolution errors before descending this positivity. Choose a Kähler form \omega_X on X, in the local smooth-potential sense, and put \Omega=\mu^*\omega_X, a smooth closed semipositive form on W. Fibre integration gives the positive constant

for general s\in S and z=\tau(s); the maps defining the birational modifications are isomorphisms along a general fibre. Let T be a closed positive current representing (6.12). The current U=l_*(T\wedge\Omega) is closed and positive, and the projection formula gives

The current on the right defining [P] is positive, closed, and supported on the finitely many curves and points in l(E'). The support theorem therefore identifies it with the integration current of an effective real divisor P on S. Furthermore,

since E' is \mu-exceptional. Hence P is \tau-exceptional. Write K_S=\tau^*K_Z+K_{S/Z}, with K_{S/Z} an actual \tau-exceptional rational divisor. Dividing the class of U by c shows that

Lemma 2.2 applies to the signed exceptional divisor c^{-1}P-K_{S/Z} and yields pseudo-effectivity of \gamma-\varepsilon\omega_Z-[K_Z]. Adding \varepsilon\omega_Z proves that \gamma-[K_Z] is big.□

Proof. The inputs are the cone theorem, projective relative vanishing and minimal-model theory, projective small factorializations and dlt modifications, and the nef-and-big Kähler criterion from [24, 22]. The higher-carrier canonical bundle formula [23, Theorem 2.3] is applied to globally gklt pairs; [23, Theorem 2.2] supplies the relative positivity used in Lemma 6.20. Each application below starts with a projective morphism.

The big non-klt contraction. Apply the construction of [24, 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 6.4 then supplies precisely the relative ampleness needed in place of [24, Lemma 2.42], including the common-resolution step in its Claim 3.5. The exceptional correction -\varepsilon[E] makes the positive part \eta^{\prime\prime} 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 [24, 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 [24, 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 [6, Lemma 8.7] gives \alpha=h^*\gamma. The descended adjoint is gklt, nef and big, and has no trivial curve. The criterion [22, 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 [4, Theorem 3.1, Step 1] applies. Its subsequent relative program starts with this projective map. Thus no general singular-source projectivity assertion is needed for this application of the criterion.

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 [39]. 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 [4, Theorem 3.1, Step 1] is needed: pullback identifies the holomorphic two-forms since the general fibre is rationally connected. The same rational Hodge correction identifies any (1,1)-class on U, modulo a class pulled back from Z, with an actual real line bundle.

Write, as in [24, 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 6.8. 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 6.6 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.

For the ample-model comparison below, make the following additional choices before running the program. Take the fixed finite cover in the form W_\ell\Subset Y_\ell\subset Z, where Y_\ell is a Stein coordinate chart and W_\ell is a closed coordinate polydisc, with the interiors of the W_\ell covering Z. Each W_\ell is a semianalytic Stein compact, so it satisfies condition (P4) of [15]; thus condition (P) holds for the restriction of every normal projective model over Z to Y_\ell.

Write

for the real line bundles N_i transported in Proposition 6.8; put N_i=0 if \alpha_U=0. These bundles are nef over Z and pull back from each contraction target, on both sides of a flip. Set a_j=a_0+j for j=1,2. The real line bundles L+jN_U are relatively ample. On each initial chart choose, once and for all,

Such representatives are obtained from general sufficiently divisible relative sections; their boundaries are relatively big because L+jN_U is relatively ample and \Delta_U\geq0. With \Xi_{0,U}:=\Delta_U+\Theta, these are actual real linear equivalences

on each chart, separately from the Bott–Chern comparison with the fixed base classes.

Let \Xi_{j,i} denote their birational transforms. Pushforward of these linear equivalences and the actual descent of N_i give

at every step. Thus the contracted adjoints have the same degrees on the negative side, and the flipped adjoints have the same degrees on the positive side. Every step of the chosen program is therefore also a (K_{U_i}+\Xi_{j,i})-negative step. Ordinary discrepancy negativity preserves klt for these fixed pairs. Their boundaries remain effective, and relative bigness is preserved by these birational pushforwards, as in the proof of [15, Lemma 11.16].

Fix these ordinary pairs on the initial charts. Their actual linear equivalences push forward throughout the program. On a common resolution at each step, exceptional negativity preserves the Bott–Chern comparison with D_U(a_0) modulo the fixed base class. Thus every subsequent contraction or flip is a step for these fixed ordinary pairs. The projective ordinary theorems [15, 17] apply. In particular, the finiteness of weak log canonical models in [15, Theorem E] does not require local \Q-factoriality of each intermediate model. Applied on the finite chart cover, its chamber argument gives termination of the global projective program with scaling. Each small step is constructed by its global rational detector algebra as above; uniqueness of the relatively ample model identifies the local ordinary flips on overlaps. This supplies a good minimal model U\dasharrow V over Z. Here one uses the projective case of [24, Proposition 2.30], not its general part (3).

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. For j=1,2, the ordinary adjoint K_V+\Xi_{j,V} is \R-Cartier and represents

It is consequently nef over every W_\ell. The pair (V,\Xi_{j,V}) is klt and its effective boundary is relatively big. Apply [15, Lemma 11.15] with A=\Xi_{j,V} and B=0. Its condition on non-klt centres is vacuous. After shrinking around W_\ell, it gives

In particular this pair is log canonical, supplies the klt boundary required in [15, Theorem 8.3], and its adjoint is \R-Cartier and nef over W_\ell. That theorem makes K_V+\Xi_{j,V} semiample on a neighbourhood of W_\ell. The pushed-forward actual linear equivalences identify these local adjoints with restrictions of the same global real line-bundle data. Their ample models therefore agree on overlaps by uniqueness and glue as above. Hence D_V(a_1) and D_V(a_2) have projective relative ample-model contractions 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. The argument of [24, Theorem 4.1, Claim 4.2], using projective relative semipositivity, gives \dim S<d. The class D_V is the adjoint of a gklt generalized pair with effective trace boundary. Its nef datum on a common carrier is the pullback of \mathbf M_U+a\alpha_U, hence is nef and big. The ordinary relative steps preserve the generalized klt condition by their negative discrepancy comparison, and preserve the nef b-datum. The space V is globally strongly \Q-factorial. Let F_V be the effective transform of F_U and put c=a+1. On a common resolution p:W\to U, q:W\to V,

where L is q-exceptional. The divisor p^*F_U-L-q^*F_V is exceptional over V and numerically trivial over V; applying negativity in both directions makes it zero. Consequently D_V=c\alpha_V+[F_V], with F_V real Cartier. Put D_S=\lambda_1, so D_V=g^*D_S is the exact Bott–Chern pullback furnished by the relative ample model. The class \alpha_S is nef by descent through the surjective morphism g.

Lemma 6.17 shows that D_V-c_1(K_V) is big. Apply its fixed-adjoint replacement to the pair on V, and choose a Kähler class \omega_S and \varepsilon>0 so small that the pullback of \varepsilon\omega_S can be subtracted from the Kähler nef datum. The resulting pair is globally gklt, its trace boundary is effective, and its adjoint is g^*(D_S-\varepsilon\omega_S). On a projective log resolution r:P\to V its structure equation is

All hypotheses of [23, Theorem 2.3] now hold: g is a projective contraction of normal compact Kähler spaces, the pair is globally generalized klt, and the adjoint equality is a Bott–Chern pullback. The theorem gives generalized klt data on S with effective trace boundary and adjoint exactly D_S-\varepsilon\omega_S. On a sufficiently high smooth carrier u:S_1\to S these satisfy

The subboundary B_{S_1} can have negative coefficients. Add \varepsilon\overline{\omega_S} to the nef b-datum. This restores the exact adjoint D_S and preserves every boundary and discrepancy. Its carrier trace \mathbf N_{S_1}+\varepsilon u^*\omega_S is nef and big. Consequently the effective trace boundary plus nef trace on S is modified big: lift the effective boundary by strict transform to a common carrier and push down the resulting big class. The nef datum throughout is a datum on the carrier.

Take a projective small strong \Q-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: smallness preserves the effective boundary trace, and on a common carrier the nef datum is still nef and big. Lemma 6.12 first gives N_\sigma(D_V)=F_V. Lemma 6.15 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 extra relative program from Claim 4.4 of the cited proof is needed.

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

Apply \mathsf G_{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 holomorphic descent and the Moishezon witness are those of Proposition 6.16.

Selected good models in dimensions at most three. Lemma 6.11 gives \mathsf G_d from \mathsf B_d and ordinary rational termination. Its smooth input has the form

where E_\mu is effective with all \mu-exceptional divisors in its support. The ordinary (K_U+B_q)-program with scaling of H is stopped at the first threshold at most one. Every performed step is negative for the displayed adjoint. Termination is the ordinary specialization of Theorem 10.6, with zero nef b-divisor. Its proof uses the projective relative and discrepancy arguments of Section 10, not \mathsf B_d or \mathsf G_d. The final nef adjoint is semiample by \mathsf B_d. Negativity removes the added exceptional divisors and gives the claimed model of X. All forward transforms used here are those of the detected ordinary steps; no inverse identification of Bott–Chern groups is needed.

The non-big non-klt contraction in dimensions at most three. Assume \mathsf B_d, \mathsf C_{d-1} and termination of existing ordinary rational klt programs in dimensions at most d, where 2\leq d\leq3. The big case having been proved, let \alpha be non-big. Take a projective dlt modification and a projective small strong factorialization, as in [24, Theorems 2.26 and 2.27]. Denote their composite by p:T\to X and use the full crepant generalized boundary on T. The underlying space T is klt, compact Kähler and globally strongly \Q-factorial. The crepant boundary is effective: its nonexceptional coefficients are the original ones, and each extracted prime has log discrepancy at most zero. The pulled-back class \alpha_T remains nef, weakly NQC and non-big. Lemma 6.19 transports the EMC hypothesis on the multiplier-ideal closed subspace to T.

We verify the bigness needed for the ordinary program. On T put

The modified-bigness resolution lemma [24, Lemma 2.10] gives bigness after adding an effective exceptional divisor E whose support contains all modification exceptional primes. Each of these primes already occurs in B_T with positive coefficient. Since \mathbf M_T is pseudo-effective by Lemma 6.17, for sufficiently small s>0 the identity

expresses Q_T as a big class plus a pseudo-effective class. For the small factorialization alone the exceptional divisor is empty. Hence Q_T is big.

Lemma 6.10 gives a finite ordinary K-negative, \alpha_T-trivial program ending in a projective Mori fibre contraction

Its class descends in Bott–Chern cohomology: \alpha_Y=f^*\gamma, with \gamma nef and weakly NQC on the normal compact Kähler base. Transport the generalized nef b-datum and the full crepant boundary along the program. On common resolutions the adjoints and structure subboundaries agree; since no step extracts a divisor, the boundary remains effective. The ordinary discrepancy comparison gives

so Q_Y=\alpha_Y-c_1(K_Y) remains big. Lemmas 6.18 and 6.19 transport both the EMC hypothesis on the multiplier-ideal scheme and, once obtained, the EMC conclusion across these projective comparisons. It therefore suffices to prove EMC for \gamma on Z.

If \dim Z\leq1, this follows from degree on a normal compact curve, using the identity for positive degree and the map to a point for degree zero. The only remaining case is \dim Y=3, \dim Z=2. The auxiliary application of [23, Theorem 2.3] in Lemma 6.20 shows that Z is klt and globally strongly \Q-factorial.

Apply Lemma 6.17 on Y with adjoint \alpha_Y. The multiplier ideal can only increase, so its new closed subspace N is a closed subspace of the previous one. It has EMC for \alpha_Y|_N by Lemma 6.18. On a projective log resolution r:W_0\to Y the new data satisfy

Suppose first that N dominates Z. Write \mathcal J=r_*\cO_{W_0}(-\lfloor B_{W_0}\rfloor). Relative Kawamata–Viehweg vanishing for fr gives

the difference with K_{W_0}+\{B_{W_0}\} is the relatively nef and big class \mathbf M_{W_0}-(fr)^*\gamma. The low-degree Leray sequence implies R^1f_*\mathcal J=0. The exact sequence of the actual multiplier-ideal scheme therefore gives a surjection

It is injective because Z is reduced and N dominates it. Thus N\to Z is a projective contraction, including on structure sheaves. EMC on N descends to Z by Lemma 6.18.

Suppose next that N does not dominate Z. The pair is gklt over a nonempty open subset of Z. Lemma 6.20, applied to these data, gives

If \gamma is not big, apply Lemma 6.10 in dimension two to (Z,\gamma). Its klt, factoriality, nefness, NQC and bigness hypotheses have all been verified; \mathsf B_2 and termination of existing ordinary surface programs are available. Its class-trivial Mori contraction has base of dimension at most one. EMC there, followed by the projective comparisons of Lemma 6.18, gives EMC on Z.

Finally suppose \gamma is big. Take a projective resolution \tau:S\to Z with S smooth and compact Kähler. The class \tau^*\gamma is big and nef. Its null curves are finite: a Kähler current with analytic singularities in this class has positive degree on every curve not contained in its singular locus. Their intersection matrix is negative definite. Indeed Hodge index is negative definite on the orthogonal complement of \tau^*\gamma, whose square is positive. The null-curve classes are independent: a relation can be written as equality of classes of two effective real divisors with disjoint component supports; their mutual intersection is nonnegative, contradicting negative definiteness unless the common class is zero, which its positive Kähler degree excludes. Grauert’s contraction criterion [21] gives a contraction of precisely these curves onto a normal compact complex surface. The contraction is projective: solve using their negative-definite matrix for an integral combination having positive degree on every contracted curve, and apply fibrewise relative ampleness. This is EMC on S, and it descends to Z by Lemma 6.18. Thus \mathsf C_d holds in all required dimensions.

Order of the induction. The assertions in dimensions zero and one are immediate from the degree of a class on a compact curve (and the ordinary curve minimal model program). For d=2,3, prove successively

At d=4 prove only \mathsf C_{4,\mathrm{big}} and \mathsf B_4. The ordinary rational termination invoked at the intermediate stages has dimension at most three. This closes the induction without using [24, Theorems 1.5 or 5.15].□

Proof. Take the fixed nef b-divisor in [38] to be zero, represented on any projective log resolution of the initial space. Its class is nef. Klt implies log canonical; the boundary is rational and is transported by strict transform. All the remaining conditions in that theorem are exactly the hypotheses just stated. No pseudo-effectivity or scaling parameter is needed.□

Proof. Write \kappa=\alpha-c_1(D) and regard its pullbacks as a nef b-class. The generalized pair (T,B+\overline\kappa) is gklt: the nef datum descends on T, so its discrepancies are exactly those of (T,B). Its boundary-plus-nef class [B]+\kappa is big. The assertion \mathsf B_d, proved in Proposition 6.21, gives f and \gamma. This assertion applies to gklt spaces without a strong factoriality hypothesis, as explained in that induction by a projective small factorialization followed by descent. Lemma 6.1, with detector D, makes -D relatively ample. The equality of the null cone with R is exactly the required contraction property.□

Proof. Start on the given pair. If its adjoint is not nef, choose a negative extremal ray. Proposition 6.23 supplies its contraction; Lemma 6.1 supplies the global relative polarization from the ordinary rational adjoint. The remaining proof of Proposition 3.4 gives the rational flip when the contraction is small, and preserves klt singularities, the global factoriality category, canonical data and the negative-side numerical ranks. These steps use the projective analytic ordinary flip theorem [15] and actual line-bundle descent. In the globally Weil case this is the argument of Proposition 3.4: the rational adjoint itself gives the global flip algebra, and the transform of each global prime is made rationally Cartier by adjusting it by a rational multiple of that adjoint and applying integral descent. The same argument applied to global rank-one reflexive sheaves preserves the strong category. Relative vanishing and cohomological descent give the one-dimensional negative-side quotient. The argument can be repeated after each finite prefix.

Let c_3(T) be the dimension of the span in H_6(T,\R) of compact analytic three-dimensional cycle classes. It is finite. A small transformation preserves it, while a divisorial contraction strictly decreases it: an exceptional divisor has nonzero class detected by the cube of a Kähler form, and its image has dimension at most two. The Borel–Moore localization argument is Lemma 4.1. There are therefore only finitely many divisorial steps in a run. An infinite run would have an infinite flip tail, contrary to Corollary 6.22, or its intrinsic formulation above. Thus the run is finite.

A maximal run ends with a nef adjoint or a Mori fibre contraction, since each negative ray otherwise admits continuation. Across each birational step the comparison on a common resolution has the form

with E exceptional over T'. Positive-current pullback and exceptional descent preserve pseudo-effectivity. A relatively antiample adjoint on a Mori fibre space is not pseudo-effective: restrict a putative positive current to a fibre curve through a point where a local potential is finite, and its nonnegative degree contradicts relative antiampleness. A nef adjoint is pseudo-effective. These observations distinguish the two endpoint cases exactly as in Section 7. Finally, the effective discrepancy comparisons add along the finite run, with strict increase at each contracted original divisor. They give the stated minimal-model discrepancy inequalities.□

Proof. Set D=K_T+B and D'=K_{T'}+B'. The ordinary comparison in Proposition 3.4 and Lemma 2.3 gives, on a common smooth compact Kähler resolution p:V\to T, q:V\to T',

where F is an actual divisor exceptional over T'. If D' is pseudo-effective, its pullback plus [F] is pseudo-effective; descent along p shows that D is pseudo-effective. Conversely, if D is pseudo-effective, exceptional descent along q applied to q^*D'+[F] proves that D' is pseudo-effective. Both uses of pullback and descent are justified by Lemma 2.2.□

Proof. Suppose that c_1(D) contains a closed positive current T with local plurisubharmonic potentials. Choose a point y where such a potential is finite and which lies on a positive-dimensional component of a fibre of g. This is possible on the dense open locus of fibres of the expected dimension: the pole set of a plurisubharmonic potential contains no open subset. That fibre component is projective, so hyperplane sections through y give an integral curve C through y contained in the fibre. Let \nu:C^\nu\to Y denote its normalization followed by inclusion.

Local potentials pull back under \nu to subharmonic functions or to the constant -\infty. Finiteness at a preimage of y excludes the latter alternative on the connected curve. Thus they define a positive current \nu^*T in the class \nu^*c_1(D), and

This contradicts D\cdot C<0, which follows from relative ampleness. The choice of y is essential: no restriction of T to a curve contained in its pole set has been used.□

Proof of Theorem 1.2. Choose a maximal ordinary negative-ray program on the given pair (X,\Delta). Proposition 3.4 supplies each step, and Theorem 6.24 proves that the program stops after finitely many such steps. Write its endpoint as (Y,\Gamma). All intermediate varieties are normal compact Kähler fourfolds, all boundaries are the rational transforms of \Delta, and all pairs remain ordinary klt and globally Weil-divisor \Q-factorial with the stipulated canonical datum. The optional strong global property is preserved by the same Proposition. The auxiliary constructions in Section 6 do not change this program, which starts on X itself.

The stopping rule gives either a nef adjoint K_Y+\Gamma or a Mori contraction g:Y\to S. In the latter case Proposition 3.4 supplies a normal compact Kähler base, a projective surjective morphism with connected fibres, \dim S<4, relative numerical rank \rho(Y/S)=1, and relative ampleness of -(K_Y+\Gamma). These are precisely the required Mori fibre space conditions, with relative rank computed from global Cartier classes. The same Proposition gives relative Bott–Chern dimension one for every negative contraction, including g.

By Lemma 7.1, the adjoint at every stage has the same pseudo-effectivity status as K_X+\Delta. If the latter is pseudo-effective, Lemma 7.2 excludes the Mori alternative, so the endpoint is nef. If it is not pseudo-effective, the endpoint cannot be nef, since nef classes are pseudo-effective by Lemma 2.2; hence the endpoint has the asserted Mori fibre structure.

For completeness, the nef endpoint has the discrepancy properties of a log minimal model. No step extracts a divisor. On a common resolution of the finite sequence, with maps p:V\to X and q:V\to Y, the ordinary effective comparison divisors add to

with F exceptional over Y. Consequently, for every prime divisor E over these models,

If a prime divisor on X is contracted, its discrepancy increases strictly at the divisorial step that contracts its transform and never decreases thereafter. Thus the displayed inequality is strict for every such prime. Here a denotes the log discrepancy, as throughout the paper.

If the initial adjoint is nef, the construction takes no steps; conversely, a zero-step nef outcome requires the initial adjoint to be nef. In the non-pseudo-effective case a zero-step program is also allowed when the first selected contraction already gives a Mori fibre structure on X. This completes both endpoint assertions.□

Proof. Define the crepant subboundary \Gamma_0 by K_Z+\Gamma_0=p^*D, using the canonical identification over the locus where p is an isomorphism. Local canonical generators and their Jacobians define this equality without choosing a global canonical divisor. Klt singularities give \operatorname{coeff}_P\Gamma_0<1 for every prime divisor P on Z. The nonexceptional coefficients are those of B and are nonnegative. Set

Both supports lie in the SNC divisor of the log resolution. This proves (8.1). A positive singular metric on a Cartier multiple of D pulls back under the dominant map p; multiplying by the divisor metric of the effective correction proves pseudo-effectivity upstairs.

Take m clearing the Cartier indices and all coefficients in (8.1). The image of \Supp E has codimension at least two in the normal space X, and p_*\cO_Z(mE)=\cO_X. Indeed, a local section of \cO_Z(mE) is a meromorphic function with poles allowed only along E. It gives a holomorphic function off that codimension-two image, which extends by normality; its pullback agrees with the original section on a dense open set. The reverse inclusion is immediate. The actual line-bundle identity

and the projection formula give (8.2).□

Proof. Restriction of the metric. Remove the critical values of g and the proper images on which a boundary stratum fails to restrict smoothly. The remaining nonempty open set T^\circ\subset T is connected. On it the fibers are smooth, \Gamma|_{F_t} is an effective SNC boundary with coefficients less than one, and adjunction is an identity of actual rational line bundles:

Here the constant one-dimensional factor coming from K_T|_t is immaterial to the line-bundle isomorphism on F_t.

Choose r>0 so that L=\cO_Z(r(K_Z+\Gamma)) is a line bundle. Give L a singular Hermitian metric of nonnegative curvature. Its local weights are plurisubharmonic. In product coordinates for the submersion over T^\circ, their local integrability and Fubini’s theorem show that the weights restrict to plurisubharmonic functions, rather than identically -\infty, on almost every fiber. A countable cover gives this assertion simultaneously for the local weights; their transition identities are preserved under restriction. Thus L|_{F_t} is analytically pseudo-effective for almost every t\in T^\circ.

Projective nonvanishing on the fibers. For almost every such parameter the fiber is also projective: the exceptions to the very-general projectivity assumption have measure zero. On a smooth projective manifold the analytic pseudo-effective cone restricted to divisor classes is the closure of the effective divisor cone, by [2, Proposition 1.2]. Hence the projective klt pair (F_t,\Gamma|_{F_t}) has pseudo-effective adjoint. The projective log MMP in dimensions at most three produces a nef model. For example, in dimension three the ordinary cone and flip theorems, together with [3, Theorem 1.2] with zero nef part, give termination; pseudo-effectivity excludes a Mori fiber space. The projective log abundance theorem [30, Theorem 1.1], together with [34, §6.1], gives a nonzero section of a multiple of the nef adjoint. Dimensions one and two use the classical log abundance theorem. The effective pullback comparison on a common resolution pulls this section back to a section of a multiple of the original adjoint. It descends to F_t by the projection formula for the modification. Consequently

for almost every t\in T^\circ. If the fiber is a point this assertion holds directly.

A common positive degree. To obtain the required quantifier, put

These are closed analytic jumping loci: the restricted family is proper and smooth, and L^{\otimes m} is locally free and flat over its base, so proper holomorphic cohomology and base change apply [11, §8, Theorem 3, p. 62]. By (8.4), their countable union contains almost every point of T^\circ. If every A_m were proper, that union would have measure zero. Therefore some A_m equals T^\circ, since this base is connected. This proves both assertions. In particular, almost-everywhere restriction of a metric has not been identified with the very-general assertion by definition.□

Proof. Proper direct image preserves coherence. On the isomorphism locus its rank is one, and double dual gives a rank-one reflexive sheaf on the normal target. Locally on T, choose meromorphic generators of this coherent sheaf and of a cleared invertible power. Pulling back their comparison gives a meromorphic comparison with the bundle upstairs. One can see that the comparison is meromorphic, rather than merely defined on a punctured open set, by using a local finite presentation of p_*L: its generators evaluate to holomorphic sections of L, and their ratios supply local denominators. The comparisons agree on overlaps because they agree on the isomorphism locus. Their divisor is exceptional. This also proves compatibility with powers and pullback. A global meromorphic frame was not used.

For a common resolution W with maps to the two spaces, first pull back the sheaf, remove torsion, and take its double dual; after an additional projective principalization it is a line bundle. Proper direct image to the other space followed by double dual is coherent. On a common codimension-one open set it is the required transform. Normality gives uniqueness of a reflexive extension across codimension at least two, proving the assertion for small maps. The additive trace identity is checked at every codimension-one point of T, where the modification is an isomorphism, and then extended reflexively. The same observation applies to canonical sheaves and their pullbacks. It requires compatible local canonical forms only.

Global strong \Q-factoriality makes M_T rationally invertible, so the first assertion defines F_M. Its negative is p-nef and has zero pushforward. Lemma 2.3 gives F_M\geq0. The ordinary crepant boundary is B_W-F_M: (9.1) gives explicitly

Thus ordinary log discrepancies are at least the generalized ones. Every ordinary zero center is a generalized zero center and is therefore generically in the specified simple normal crossing open set. This proves the klt and dlt assertions.

The effective divisor \lfloor B\rfloor is rational Cartier on T. Decreasing it subtracts the effective divisor \epsilon p^*\lfloor B\rfloor from the crepant boundary. A former positive discrepancy stays positive. Every former zero-discrepancy place has center in the floor, since its center is generically a coefficient-one stratum, and hence has strictly positive order on the pullback of the floor. Its new discrepancy is positive as well. The boundary on T remains effective for 0<\epsilon<1.□

Proof. Write p\colon W\to T for a smooth carrier, with p projective, and put \rho=\pi p. Work on a Stein neighborhood U of the prescribed compact. Choose a \rho-ample rational line bundle A. For an integer m clearing denominators, the sheaf

is coherent and has rank one over the bimeromorphic isomorphism locus. It is nonzero. Cartan’s Theorem A gives a nonzero section over U, whose pullback defines an effective divisor. Hence, after division by m,

This proves relative bigness directly, including when M_W=0. The divisor E need not be exceptional. The equivalence comes from a section of the specified bundle, and introduces no unspecified numerical or base twist.

Keep W and the \rho-ample bundle A fixed. Choose an auxiliary log resolution of the finite fixed supports only to test the singularities. On that auxiliary resolution there are finitely many crepant coefficients over the compact preimage, all strictly below one. Hence a sufficiently small rational \epsilon>0 makes (W,B_W+\epsilon E) sub-klt. The rational bundle

is \rho-ample. Choose a rational \eta>0 such that A_\epsilon-\eta p^*H is still \rho-ample. On a smaller neighborhood, sufficiently large powers of these relatively ample bundles are relatively generated. Cartan’s Theorem A supplies finitely many sections generating over the compact preimage; shrink once more so that they generate throughout. Divided general members give

Choose the members on T and W, respectively, with large denominators, and test generality on the auxiliary log resolution. Their pulled-back systems remain free; we do not claim that a pulled-back ample bundle stays ample. Analytic Bertini gives

sub-klt. Here generation is used for the ample systems only; the nonzero section producing E was not claimed to generate its bundle. The analytic generation and Bertini statements used in this step are [6, Theorems 2.12 and 2.20].

Push down to obtain

Only exceptional coefficients of B_W could be negative, so \Delta is effective. The actual upstairs identity is K_W+C_W\simq p^*D_T. Choose an integer clearing this identity, push forward the resulting line-bundle isomorphism, and take double duals. Agreement in codimension one gives

In particular, the new ordinary adjoint is rational Cartier. We have not inferred Cartierness of newly chosen local divisors from global strong \Q-factoriality. The divisor

is exceptional and p-numerically trivial by the same line-bundle identity. Lemma 2.3 in both signs makes it zero. Thus C_W is the ordinary crepant boundary, and (T,\Delta) is klt. Subtracting \delta H_0 gives the asserted representation of D_T-\delta H and leaves the common ample part (\eta/2)H_0.

For finitely many vertices, use one carrier and one H_0, choose their effective divisors in (9.4), and resolve all fixed supports together. Take a common sufficiently small \epsilon and then a common \eta. Choose the finitely many free members generally. On the resulting common resolution all the crepant coefficients are uniformly below one. Convex combinations are therefore klt and keep the same ample part. The parameter space can be the simplex of vertex weights; no affine independence of the original vertices is required.

For real data, first express the bundle in a finite real span of rational bundles on the carrier. Every signed rational generator has an effective representative over the Stein neighborhood by the same Cartan argument. Thus a real nef bundle also has an expression M_W\sim_{\R}A+E with A relatively ample and E\geq0 in a fixed finite divisor span. The preceding small-\epsilon construction applies. A real relatively ample bundle can be written as a positive combination of finitely many nearby rational relatively ample bundles; divided general members of those bundles give the required real boundary. The ordinary real klt cone theorem is consequently applicable. Only the rational construction is used for line-bundle descent and for the finite adjoint rings.□

Proof. For an exceptional prime E, global strong \Q-factoriality makes E rational Cartier. If it were f-nef, Lemma 2.3(i) applied to -E would give -E\geq0. Thus E\cdot R<0. Every contracted curve must lie in E, since an effective rational Cartier divisor has nonnegative intersection with a curve not contained in its support. Every point of a nontrivial projective fiber lies on a curve in that fiber. Hence the entire exceptional locus is E, and no second exceptional prime exists.

For a global rank-one reflexive sheaf \cF on Z, form its reflexive pullback to T. A positive power is a line bundle. Subtract a rational multiple of E so that its degree is zero on R and apply Lemma 3.1. The descended rational bundle agrees with \cF to that power on the complement of a codimension-two set in Z. Reflexivity extends the identity. This proves the global strong condition and, in particular, applies to the pushed-forward canonical, boundary, and nef-trace sheaves.

In the small case, L-cJ has zero degree and descends by Lemma 3.1. The right-hand side of (9.6) is a rational line bundle on T^+ and agrees with the transform of L on the common open set. It is therefore the reflexive transform. The sign conclusions follow from relative ampleness of J^+ and the sign of c.

Conversely, transport an arbitrary global rank-one reflexive sheaf \cG on T^+ to T as in Lemma 9.3. A reflexive power of its transform is invertible on T. The preceding argument makes its transform a rational line bundle on T^+. It agrees with the corresponding reflexive power of \cG on the common codimension-one open set, and hence everywhere. This is global strong \Q-factoriality on T^+; no assertion about arbitrary open subsets is involved.□

Proof. First assume gklt. The global cone construction above gives a negative ray and a nef rational supporting line bundle L which vanishes precisely on that ray and for which a multiple minus the adjoint is relatively ample. Apply the ordinary base point free theorem on the finite Stein cover, using Lemma 9.4. It generates a common positive power of L relative to V. The relative section morphism and its Stein factorization give f\colon T\to Z with connected fibers and normal target, contracting exactly the curves on the selected ray. The target is projective over V: it is finite over the image in the projectivization of the coherent sheaf of relative sections. Since T\to V is bimeromorphic, Z has the same dimension and f is bimeromorphic. The negative adjoint is relatively ample, because on every f-fiber its class is a positive multiple of a relatively ample class in the one-dimensional global degree space.

If f is divisorial, apply Lemma 9.5. If it is small, apply Lemma 9.4 locally over Z, followed by the ordinary analytic klt flip theorem [15, Theorem 1.14]. Equivalently, the actual relative adjoint algebra is locally finitely generated by [6, Corollary 3.7]; the flip theorem also supplies its normality and smallness. The local models glue by the intrinsic algebra. Compactness permits a common Veronese generated in degree one. On the common codimension-one open set its tautological bundle is the stated adjoint power. Both sheaves are reflexive on the normal positive side, so this identity extends globally. Thus the positive adjoint is rational Cartier and relatively ample as an actual bundle. Lemma 9.5 gives global strong \Q-factoriality.

All spaces constructed are projective over a compact Kähler space. A relatively ample line bundle has a metric with positive curvature in the fiber directions; adding a sufficiently large pullback of a Kähler form on the base gives a Kähler form. Thus the spaces are compact Kähler. The common carrier remains projective over the models: take the main component of the graph over Z and resolve it. The pulled-back nef data are still nef over V.

For gdlt data, replace B by B-\epsilon\lfloor B\rfloor with small rational \epsilon>0. By Lemma 9.3 this is gklt; denote its adjoint by D_\epsilon. Normalize by a relatively ample bundle H, and choose a relative curve class z with H\cdot z=1 and D\cdot z=-a<0. Choose \epsilon so that |(D_\epsilon-D)\cdot u|<a/4 on the whole compact normalized slice. Choose an extreme point u_0 of the face where D attains its minimum on this slice. It is an extreme point of the entire slice, so it spans a global extremal ray R. Since D\cdot u_0\leq D\cdot z=-a, the uniform perturbation bound gives D_\epsilon\cdot u_0<-3a/4. Thus R is negative for both adjoints. The gklt construction contracts this global extremal ray for the perturbed pair. The original adjoint has the same negative sign on the ray. In the small case Lemma 9.5 identifies its positive transform with a positive multiple of the perturbed positive adjoint modulo a base pullback. In the divisorial case that lemma makes its pushdown rational Cartier. These are therefore steps for the original data.

Finally Lemma 2.3 proves the discrepancy comparisons. For gklt, positivity is preserved. For gdlt, a new zero-discrepancy center cannot have its general point in an exceptional locus, by strictness. Its corresponding old zero center is generically in the unchanged good open set. The transformed boundary is SNC and the b-data descend there. This gives a gdlt good open set on the output. The same hypotheses hold for the next relative morphism to V, so the construction continues whenever relative nefness fails.□

Proof. Lemma 9.4 supplies ordinary klt representatives with a common positive relatively ample part. Choose a common projective log resolution a\colon S\to T by successive blowups and principalizations with centers exceptional over T, resolving all fixed supports. This choice can be made with an effective exceptional divisor F such that -F is a-ample: for each blowup take its effective tautological exceptional divisor, and inductively add it to a sufficiently large positive multiple of the pullback of the divisor for the preceding composite. Relative ampleness under composition proves the assertion; further such blowups preserve it. For the identity resolution take F=0. This is the blowup-resolution construction used in [6, Theorem 2.13, Remark 2.14, and the proof of Corollary 3.7]. For each vertex choose every exceptional coefficient strictly between the maximum of zero and its crepant coefficient and one. We obtain

All supports, including that of F, are SNC. The ample part downstairs permits a common relatively ample part upstairs. More explicitly, scale the chosen F by a small positive rational number so that a^*H-F is ample over V, where H is the fixed ample part downstairs. For a sufficiently small \tau>0 put A=\tau(a^*H-F). The finitely many exceptional coefficients of the \Gamma_j have positive margins from both zero and one. Then \Gamma_j-A is effective with coefficients below one; their supports, including that of F, are SNC. One can either use this relatively ample rational divisor A directly or replace it by a divided general member and make the corresponding simultaneous linear-equivalence changes in the boundaries.

The multigraded ring of the K_S+\Gamma_j is locally finitely generated by [6, Theorem 3.1]. For nonnegative integers n_j, the correction \sum n_jE_j is effective and exceptional over T. Normality gives a_*\cO_S(\sum n_jE_j)=\cO_T. The projection formula therefore identifies every multigraded piece with the corresponding piece for the D_j, compatibly with multiplication. Passing to one common Veronese removes all rounding. Thus their actual multigraded ring is locally finitely generated as well.

For completeness, finite generation gives the required finiteness as follows. On a sufficiently small neighborhood choose homogeneous generators with multidegrees d_1,\ldots,d_N\in\N^{s+1}. For a rational weight w in the simplex, a point of the relative spectrum is semistable for the diagonal ring of weight w precisely when w is in the cone generated by those d_i whose generators are nonzero at the point. To verify this equivalence, clear the rational coefficients of such a cone expression and form the corresponding monomial; conversely a nonvanishing homogeneous section contains a nonvanishing monomial of that weight. There are only finitely many subsets of the generators and hence finitely many resulting open sets. The diagonal Proj is the quotient of its open set, obtained by the degree-zero localizations of these monomials. Uniqueness of this quotient shows that weights with the same open set give the same ample model. All identifications commute with the map from T, which is determined on the bimeromorphic isomorphism locus. A finite cover of the Stein compact makes the number of possibilities finite there. This is also the marked finite-model conclusion of [15, Theorem E], now with the ordinary representatives and the smooth reduction justified.□

Proof. Let D=K_T+B+M_T. We may suppose D is not already relatively nef.

A scaling class valid on every later model. Choose global rational line bundles L_1,\ldots,L_r whose degrees form a basis of N^1_{\mathrm{gl}}(T/V)_{\Q}, and write the class of D as d\in\Q^r.

These bundles continue to span the degree space after every finite sequence of elementary steps. Indeed every line bundle on a later model can be transported to a rational line bundle on the preceding one by Lemma 9.3 and global strong \Q-factoriality. For a divisorial contraction its pullback is also such a bundle; exceptional-divisor adjustments are available on the source. Moreover a bundle numerically trivial over V remains so after a step. It first descends across that contraction by Lemma 3.1. Any curve on the new side over V has a projective lift to a common model and then a curve upstairs mapping onto it with positive degree. Intersecting the pullbacks of the descended bundle proves its zero degree. Thus every later relative curve determines a rational functional v\in\Q^r, nonzero because the later space has a relatively ample line bundle.

Choose a real vector h\in\R^r defining a real linear combination H of the chosen rational line bundles, with H and D+H relatively ample. We choose it outside the following countable family of proper rational hyperplanes:

The ample conditions define a nonempty open set, and a countable union of proper hyperplanes cannot exhaust it. The rational functionals here range over all possibilities, so this single choice works at every later model; no enumeration of the models is required.

The scaling steps and their strictly decreasing thresholds. On a current model, let D_i be the reflexive transform of D, and let H_i be the same real linear combination of the reflexive transforms of the rational line bundles defining H; across a divisorial step these transforms are the corresponding pushforwards. Regard H also as nef b-data over V, carried on the initial space. Its pullback to a common carrier is relatively nef, so for every t\geq0 the initial generalized pair with adjoint D+tH is gklt. At each preceding step of threshold t_j, its adjoint has negative degree for 0\leq t<t_j and zero degree for t=t_j. Lemma 2.3 consequently shows that the transformed pair is gklt throughout the parameter interval below the preceding threshold.

Start with the relatively ample class D+H. On a current model take the lower endpoint t_i of the nef segment extending downward from the preceding parameter. If t_i=0, then D_i is nef and the process stops. Otherwise J_i=D_i+t_iH_i is nef and not ample. The local cone argument preceding Lemma 9.5, applied to this real gklt adjoint, shows that its zero face is generated by finitely many curves. A generating curve cannot have D_i-degree zero, because then its H_i-degree would be zero, contrary to (9.7). Two independent such rays would violate (9.8). Hence the face is one ray in the global degree space. Its D_i-degree is negative: if it were positive, lowering t would be positive on this face, and the uniform cone margin would give nefness below t_i, a contradiction.

Apply Proposition 9.7 to this D_i-negative ray. The threshold class descends. Here descent for a real class requires no real-bundle base point free theorem: the perpendicular hyperplane of the rational ray is rational, so express the threshold class as a real combination of rational line-bundle classes of degree zero and descend each by Lemma 3.1. The threshold comparison is crepant and its transform is nef.

The thresholds strictly decrease. In a small step, the zero face on the positive side is again generated by curves by the real gklt local cone argument. Equations (9.7)– (9.8) allow at most one ray. Flipped curves supply that ray and have positive D-degree. Lowering the parameter is therefore positive on it, and the uniform margin makes the class ample for a nonempty interval below the old threshold. In a divisorial step there are no zero curves at the old threshold: lifting one to the old space would make it lie on the contracted zero ray while still mapping onto a curve. The same margin again gives an ample interval. At every stage, the open interval between successive thresholds consists of relatively ample classes: an interior zero curve would have zero degree for two parameters, hence zero H_i-degree, contrary to (9.7).

Finitely many marked ample models. It remains to exclude infinitely many such intervals. Choose finitely many rational relatively ample bundles H^{(1)},\ldots,H^{(s)} near H, with H in the interior of their convex hull and with D+H^{(j)} relatively ample. Consider the fixed rational simplex of weights with vertices

At an interior parameter of any one of the alleged infinitely many ample intervals, all preceding intersections for that parameter are strictly negative and final ampleness is open. Choose a nearby rational point of (9.9) retaining these finitely many strict conditions. The current model is its marked ample model: on a common resolution its pullback difference from the initial adjoint is effective and exceptional over the current model by composition of the preceding comparisons. The corresponding relative section algebras agree, since pushing an effective exceptional correction to a normal target gives the structure sheaf.

Each vertex in (9.9) has nef b-data on the initial model and is gklt. Apply Lemma 9.8 on each member of a finite Stein compact cover of V. It gives finitely many local marked ample models, hence finitely many possible tuples of such local models. Two global models giving the same tuple are uniquely isomorphic over the interiors of these compacts: the identification is fixed on the common bimeromorphic open set and extends by the marked local isomorphism. The identifications agree on overlaps by normality, so they glue. Thus only finitely many global marked ample models occur.

A marked model cannot recur across a nontrivial intervening step. For a fixed marking T\dashrightarrow Y, choose a common resolution. The pullback difference for D+tH is affine in t. Its coefficient inequalities of effectivity are convex, and the relatively ample cone of Y is convex. The parameters for which Y is its marked ample model therefore form an interval. If it occurred in two separated ample intervals of the run, it would also be the ample model at every intermediate ample parameter. Uniqueness of the relative Proj would identify each intermediate model with the same marking. But adjacent models across a nontrivial elementary step have different markings, as their adjoint has opposite relative signs. This is impossible. Finitely many marked models therefore imply finitely many steps.□

Proof. The projective realization statement in Lemma 4.2 places the finitely many designated divisors on a projective log resolution carrying the nef data. Its proof uses only log-smooth discrepancy calculations and stratum blowups, independently of extraction. Write this resolution as p\colon W\to T. Keep every designated coefficient at its crepant value 1-a(E), which lies in [0,1). Keep the nonexceptional coefficients as well. For every other exceptional prime choose a coefficient in [0,1) strictly greater than its crepant coefficient. Since the pair is gklt, there are only finitely many such choices and they are possible. With this effective SNC boundary \Gamma we have

where the support of G is exactly the set of undesired exceptional primes. The new pair is gklt.

Apply Proposition 9.9 over T. At every stage the adjoint is the pullback of D_T plus the effective transform of G, and the latter remains exceptional over T. A divisorial negative step cannot contract a prime outside its support: a general covering contracted curve in that prime avoids containment in the support and has nonnegative intersection with the effective rational Cartier divisor G, whereas its adjoint degree is negative. Small steps contract no prime divisor. Hence every designated prime survives, and no nonexceptional prime of the resolution is lost.

At the relative nef endpoint h\colon U\to T, the remaining effective exceptional correction G_U is h-nef. Negativity gives G_U\leq0, hence G_U=0. Every undesired prime has therefore been contracted. The equality is now crepant, and the retained boundary is effective. The global strong condition, Kählerness, and gklt property follow at each step from Proposition 9.7. The exceptional primes are exactly the specified ones, as claimed.□

Proof. Write the sequence as

The boundary is transformed strictly, so its coefficients belong to one finite set. Smallness and Lemma 2.3 preserve generalized terminality. Forgetting the effective boundary and the effective nef defect increases discrepancies, so each X_i is an ordinary terminal threefold. In particular its singularities are isolated, and the general point of every flipped curve is smooth.

Choose an integer p>0 for which p\bM is represented by an integral line bundle on a carrier, and an integer N divisible by p and by the denominators of the boundary coefficients. These integers continue to work on higher carriers throughout any finite part of the sequence. If C is a flipped curve on X_{i+1}, the exceptional valuation E_C of its generic blowup has log discrepancy

Indeed, the ordinary blowup contribution is two, the m_j are the generic multiplicities of the boundary components, and the remaining term is the coefficient of the nef defect. At this smooth generic point the integral trace of p\bM is Cartier. The defect therefore has coefficients in p^{-1}\Z and is effective by Lemma 9.3. This proves (10.1), including its signs.

Let b be the largest boundary coefficient, with b=0 when the boundary is empty. For b>0 test the flipped curves contained in a coefficient-b component; for b=0 test every flipped curve. The value in (10.1) belongs to the fixed finite set

For each t\in\mathcal A_b, set

Lemma 4.2 makes these numbers finite, since the terminal cutoff is the strict inequality a<2-b. Small transformations identify the sets of exceptional valuations, and discrepancy monotonicity makes every d_t(i) nonincreasing. For the valuation of a tested curve, Lemma 2.3 gives

Thus d_t drops at that step. The nonnegative integer \sum_{t\in\mathcal A_b}d_t(i) can drop only finitely often. If b=0, every nontrivial small step has a flipped curve, so the proof is complete.

Suppose b>0. On a tail, no flipped curve is contained in any of the coefficient-b components. Fix such a component D_i, and let D_i^\nu denote its normalization. The two images of D_i and D_{i+1} in Z_i agree, because the ambient transformation is an isomorphism in codimension one. Denote the normalization of this common image by V_i. There are projective birational morphisms

The morphism on the right contracts no curve and is therefore finite: every positive-dimensional projective fiber contains a curve. A finite birational morphism to a normal space is an isomorphism. Hence there is a projective birational morphism D_i^\nu\to D_{i+1}^\nu. It contracts exactly the curves whose images in X_i are flipping curves contained in D_i. By Lemma 4.1, with k=1, each such contraction strictly decreases the dimension of the span of curve classes of the compact normalized surface. There are finitely many coefficient-b components, so only finitely many further steps have a flipping curve contained in one of them.

On the resulting tail put D_i^{\max}=\sum_{b_j=b}D_{i,j} and replace B_i by B_i-bD_i^{\max}. Every contracted curve C has D_i^{\max}\cdot C\geq0, because it is not contained in this effective \Q-Cartier divisor. Thus the new adjoint

is negative on the old ray. To check its positive transform, write A_i=K_{X_i}+B_i+M_{X_i} and choose the rational number c=(A_i'\cdot C)/(A_i\cdot C)>0. The global one-ray condition and Lemma 3.1 give an actual identity of rational line bundles

Its strict transform is A_{i+1}'=cA_{i+1}+(f_i^+)^*L_i, which is f_i^+-ample. Consequently the same tail is a sequence of elementary steps for the new generalized pair. Decreasing the effective boundary preserves generalized terminality. Induction on the number of distinct positive boundary coefficients proves the proposition.□

Proof. For the first assertion, use the analytic index-one cover \pi:(T^\sharp,x^\sharp)\to(T,x), a cyclic cover of degree r that is étale away from the distinguished point. The terminal threefold classification makes T^\sharp smooth or an isolated cDV hypersurface germ; see [37]. A sufficiently small punctured representative U^\sharp is simply connected, by the connectivity theorem for links of isolated hypersurfaces [35, Theorems 2.10 and 5.2]. For a convergent analytic defining equation, finite determinacy gives an analytically equivalent sufficiently high Taylor polynomial [8, Theorem 6.2, p. 24], so the same connectivity statement applies; the smooth case has the same property. For every m>0, the analytic Kummer sequence implies that the m-torsion of \Pic(U^\sharp) is an image of H^1(U^\sharp,\boldsymbol\mu_m)=0. The local divisor class group injects into this Picard group: a rank-one reflexive sheaf trivial on the punctured normal germ is trivial on the germ by reflexive extension. It follows that the local class group of T^\sharp is torsion-free.

Since D is \Q-Cartier, the integral divisor \pi^*D has torsion class and is therefore principal, say \pi^*D=\operatorname{div}(u) for a meromorphic germ u. The meromorphic norm gives

Thus rD is Cartier; compare the index divisibility statement in [27, Lemma 5.1]. The argument above proves the analytic assertion for divisor germs already known to be \Q-Cartier.

The second assertion is the terminal small-discrepancy theorem [28, Appendix, pp. 193–195]. Its proof uses the analytic quotient classification and gives ordinary discrepancy 1/r, or log discrepancy 1+1/r in our convention.□

Proof. Let \nu:P^\nu\to P be the normalization and let \mu:S\to P^\nu be its minimal resolution. Divisorial adjunction gives

as an identity of rational adjoint bundles. Neither normality of P nor log canonicity of (U,P) is needed for effectivity of the different in this formula. This is the ordinary different of a reduced prime, computed at codimension-two points of U; the construction and effectivity are given in [14, Section 14.1(i)–(iii), pp. 40–41], and analytic adjunction is included in [16, Theorem 1.1 and the following setup, p. 1]. Here is the local analytic effectivity argument. On a transverse klt surface T, take a small smooth quotient q:V\to T and write P_T for the reduced boundary curve and C=(q^{-1}P_T)_{\mathrm{red}}. The quotient is unramified at the general points of these curves, so q^*(K_T+P_T)=K_V+C. Let \rho:C^\nu\to P_T^\nu be the induced map of normalizations. Residue adjunction for the reduced plane curve C has an effective conductor divisor A, and the canonical ramification formula for \rho has an effective divisor R. Compatibility of residues with pullback gives

At a branch of ramification degree e, its coefficient downstairs is (c+e-1)/e\geq0, where c is the conductor coefficient. This proves the analytic effectivity in (10.2) without an lc hypothesis on (U,P) or normality of P.

The sum on the right of (10.2) is rational Cartier; its individual canonical term need not be. If that term is used separately, take its numerical pullback in the sense of Mumford, determined by intersection zero with all exceptional curves. The negative definite exceptional intersection matrix makes this pullback linear and makes the pullback of an effective divisor effective. In particular the calculation does not presume \Q-factoriality of P^\nu.

For completeness, the effective correction can also be checked directly on S. For every irreducible \mu-exceptional curve E, adjunction on the smooth surface gives

Indeed, an exceptional smooth rational (-1)-curve is excluded by minimality over P^\nu; every other exceptional curve satisfies the displayed inequality. The pullback of the adjoint in (10.2) therefore has the form

To obtain the sign, the residue comparison defines \Delta_S as a rational divisor with pushforward \Delta_{P^\nu}\geq0. Since K_S is \mu-nef, the divisor -\Delta_S is \mu-nef. Apply Lemma 2.3(i) to E=\Delta_S to obtain \Delta_S\geq0. This is equivalently the numerical-pullback calculation above, and proves (10.3) as an actual bundle comparison.

The image h(P) has dimension zero or one. In the latter case, general smooth fiber components of S\to h(P) are moving curves F with K_S\cdot F=2g(F)-2\geq-2. One may use the Stein factorization to describe these fibers over a smooth compact curve. Choose a general fiber component not contained in \Supp\Delta_S, the conductor, or the pullbacks of the D_j.

If h(P) is a point, P is a projective fiber subspace, and S is a smooth projective surface. Take a surface minimal model \rho:S\to S_0. If K_{S_0} is nef, use general ample curves on S_0. Otherwise the classification of smooth projective surfaces gives ruling fibers with canonical degree -2, or lines on \mathbb P^2 with canonical degree -3. Their general strict transforms F satisfy K_S\cdot F\geq-3, because K_S-\rho^*K_{S_0} is effective. In all cases the curves cover S, so F may be chosen not contained in the finitely many excluded divisors. This is the surface argument also used in [3, Lemma 3.3].

The curve F maps birationally to a curve C\subset P: it is not contained in the exceptional or conductor locus. By the projection formula and (10.3),

Its choice also ensures C\not\subset\Supp D_j for every j.□

Proof. We induct on the number of exceptional places with log discrepancy at most one. The terminal case is controlled by local canonical indices. For the induction step, extract one place, bound its intersection degree using the preceding surface estimate, and descend an integral line bundle.

Replace \varepsilon by \min\{\varepsilon,1\}, so assume 0<\varepsilon\leq1. We construct integers divisible by those from the preceding induction stage. When n=0, the pair is generalized terminal. The underlying space is ordinary terminal and

for every exceptional valuation. By Lemma 10.2 and the observation following that lemma, every canonical point index r>1 satisfies r\leq1/\varepsilon. Each local representative of \cF is \Q-Cartier by the global strong condition, and its Cartier index divides r. Thus one may take

Local invertibility of this reflexive power at every point is precisely global invertibility; no trivialization on all of T is asserted.

Suppose n>0 and put I'=I(n-1,\varepsilon). If there is no exceptional valuation of discrepancy at most one, the preceding case applies. Otherwise choose one such place, of discrepancy a\in[\varepsilon,1], and apply Proposition 9.11 to obtain a projective extraction h:U\to T with that single exceptional prime P. Write its effective crepant boundary as

Crepancy preserves all discrepancies and lowers the number of exceptional low places by one. The induction hypotheses therefore hold on U, so I' clears the index of every global rank-one reflexive sheaf on U.

First, -P is h-ample. If H is an h-ample line bundle, its reflexive pushforward is rational Cartier on T. Lemma 9.3 gives

as a rational bundle comparison. The divisor cP is h-ample and h-exceptional, hence is nonpositive by negativity. Since h has a positive-dimensional fiber, c\neq0, and therefore c<0.

The same trace comparison for the nef data gives

Apply Lemma 10.3 with the components of B_U^{\mathrm{str}} as the excluded divisors. For its contracted curve C\subset P, the other boundary has nonnegative degree and M_U\cdot C=-sP\cdot C\geq0. Crepancy yields

Consequently

Now let \cF be any global rank-one reflexive sheaf on T, and let \cF_U be its reflexive strict transform. In additive rational-bundle notation, Lemma 9.3 gives

This comparison is defined by local meromorphic frames agreeing off the exceptional locus, and requires no global meromorphic section of \cF. The sheaves \cF_U^{[I']} and \cO_U(I'P) are line bundles. Intersecting (10.5) with C gives

The numerator is an integer. Define

Then I:=I(n,\varepsilon) is divisible by I' and Iq\in I'\Z. Thus I\cF_U+(Iq)P represents an integral line bundle L on U, with its prescribed comparison to h^*\cF^{[I]} off the exceptional locus. Equation (10.5) shows that L has degree zero on every h-contracted curve.

Increase the coefficient 1-a of P by a sufficiently small positive rational number \delta. On a fixed log resolution carrying the data, the inequalities defining generalized klt remain strict for all sufficiently small \delta, so the new pair is effective and generalized klt. Its adjoint is

which is h-antiample. Lemma 3.1 now descends L integrally. More explicitly, local ordinary klt replacement and the analytic base-point-free theorem generate L^m for every sufficiently large integer m; descend two consecutive powers and take their quotient. No further index is introduced. The descended line bundle agrees with \cF^{[I]} away from h(P), a set of codimension at least two in T. Reflexive extension identifies them on T. This proves the induction, including the case a=1.□

Proof. Set E=P-P_+=F-G. Over Y, the divisor -E=P_+-P is nef, while

since G is p-exceptional. Lemma 2.3 gives -E\leq0, proving the first assertion. If A is nef over S, then E is nef over T_+ and

since F is q-exceptional. Negativity gives E\leq0, and hence E=0. The two exceptional supports need not coincide.

Assume A_+ is relatively ample. A fiber of p maps by q into a projective fiber of T_+\to S. Equality p^*A=q^*A_+ gives degree zero on every curve in that fiber. A nonconstant image would be detected by a curve of positive degree against A_+, so q is constant on each connected fiber of p. Proper factorization through the normal space Y gives q=g\circ p. The map g is projective because Y\to S is projective and T_+\to S is separated. Finally, pullback by p is injective on rational line bundles, by p_*\cO_W=\cO_Y and the projection formula; therefore A=g^*A_+. This proves the actual bundle identity, not merely numerical equality. Compare [32, Lemma 2.12] for the algebraic form of this comparison.□

Proof. In dimension at most two there are no nontrivial small birational contractions, and Lemma 4.1 gives the result. In dimension three that lemma first removes a finite divisorial prefix. Relabel the remaining small sequence as

All boundaries are now strict transforms of B_1.

By Lemma 4.2, there is \varepsilon_0>0 such that the initial set of exceptional valuations with discrepancy below 1+\varepsilon_0 is finite. Since discrepancies increase and small maps preserve exceptionality, the sets

form a decreasing sequence of finite sets. After discarding a finite prefix, they equal one set \mathcal L=\{E_1,\ldots,E_n\}. At the first model of this tail, the finitely many exceptional values in (1,1+\varepsilon_0) have a positive minimum distance from one; use \varepsilon_0 itself if there are no such values. Choose \varepsilon>0 no larger than this distance, than \varepsilon_0, than one, and than the positive lower bound for all discrepancies of the initial gklt pair furnished by Lemma 4.2. Monotonicity and the stabilization of \mathcal L show that, on every model of this tail, all discrepancies are at least \varepsilon, and every exceptional discrepancy greater than one is at least 1+\varepsilon.

Apply Lemma 10.4 with these fixed n and \varepsilon. Let r clear the coefficients of B_1 and let p clear the higher nef datum. For example

makes each J(K_{X_i}+B_i+M_{X_i}) an integral line bundle, and makes J\bM integral on every higher carrier. Comparing the actual adjoint bundles on a smooth carrier shows that every discrepancy belongs to J^{-1}\Z: the canonical bundle there is integral and the coefficients of the corresponding integral divisor difference are integers. The same argument applies after any further resolution. Thus each nondecreasing sequence

is eventually constant. After another finite prefix, all these values are constant simultaneously.

Use Proposition 9.11 to extract exactly E_1,\ldots,E_n over the first model of this final tail:

The boundary is effective and crepant. Every valuation exceptional over Y_1 is exceptional over X_1 and is not in \mathcal L, so its discrepancy is greater than one. Hence this is a globally strongly \Q-factorial generalized terminal model. If n=0, take h_1=\id.

Consider the first remaining flip over Z_1. Starting from Y_1, use Proposition 9.7 to take relative steps over Z_1 while the adjoint is not nef. We show that no divisorial contraction can occur in this run. At any finite stage Y_j, a common smooth model gives

where P_0 is pulled back from Y_1, P_j from the current lifted adjoint, and P_+ from K_{X_2}+B_2+M_{X_2}. These are the prescribed adjoint comparisons. The accumulated error G_j is effective and exceptional over Y_j; the error F of the original flip is effective and exceptional over X_2. The positive adjoint is ample over Z_1. The first assertion of Lemma 10.5, which does not require nefness on Y_j, gives G_j\leq F.

If a divisor were lost, its strict transform from Y_1 would have a positive coefficient in G_j, by the strict discrepancy comparison at its contraction. But every divisor on Y_1 is either the transform of a divisor on X_1, or one of the E_\ell. In the first case its coefficient in F is zero because the original map is small. In the second case that coefficient is

by stabilization. Both cases contradict G_j\leq F. Thus the entire lifted run is small. It stays generalized terminal, and Proposition 10.1 forces it to reach a nef endpoint after finitely many steps.

At that endpoint, Lemma 10.5 gives equality of the two adjoint pullbacks and a projective crepant morphism

Here Y_2 denotes the endpoint, rather than the first intermediate model of the finite run. Since the lifted transformations were small, the exceptional primes of h_2 are exactly the same E_\ell, with the same boundary coefficients. We may therefore repeat this construction over each successive base Z_i.

Each nontrivial flip below forces at least one step above. Indeed, take a negative curve C\subset X_i over Z_i. Projectivity of h_i supplies a curve \Gamma\subset Y_i mapping onto C with positive degree. The crepant identity gives

Thus the starting lifted adjoint is not nef over Z_i. An infinite original sequence would consequently concatenate to an infinite sequence of small generalized terminal elementary steps, with fixed rational boundary coefficients and the same rational nef b-line data. All spaces remain compact Kähler and globally strongly \Q-factorial. The bases may change, exactly as allowed in Proposition 10.1, which gives the contradiction.□

Proof. Choose a projective log resolution h:W\to V carrying the nef data which is an isomorphism at the general points of the good strata under consideration. We may replace W further whenever necessary, preserving these general points. Since V is globally strongly \Q-factorial, the trace M_V is a rational line bundle. Lemma 9.3 and Lemma 2.3 give

Use compatible local canonical representatives to write

These are actual rational sheaf comparisons, and A_W=A_W^{\rm o}+H.

We first justify the ordinary companion in this construction. The inequality above says that ordinary log discrepancies of (V,A) are at least the generalized log discrepancies. In particular (V,A) is lc. An ordinary zero-discrepancy place is also a generalized zero-discrepancy place. Its center therefore meets the good open set. Conversely, every generalized center meets that set, where H=0, the data descend, and the log formula is the ordinary snc formula. Such a center is an ordinary coefficient-one stratum. Thus (V,A) is ordinary dlt and its lc centers are exactly the generalized ones. This argument also explains why forgetting the nef part does not introduce an additional zero center outside the good open set; compare the algebraic formulation in [3, Remark 2.6(1)].

Let S\subset V be a prime component of \lfloor A\rfloor and let S_W be its strict transform. It is smooth after the chosen resolution. The coefficient of S_W is one in both formulas. The residue isomorphism restricts them to S_W as

The normal space S_W maps to the normalization S^\nu. Push forward the indicated boundaries and the nef trace to S^\nu. Over its smooth large open set, the residue identification is the adjunction identification of the canonical sheaf. It therefore defines, by reflexive extension, the rational adjoint bundle on S^\nu as the pullback of the ambient adjoint. This construction gives an actual line-bundle isomorphism after a common positive multiple. The prescribed residue identification, and not equality of first Chern classes, specifies the comparison.

The restricted log boundary on S_W is snc with coefficients at most one. Thus the resulting generalized pair is sub-lc before effectivity is checked. Every zero stratum of that restricted log formula is an intersection of S_W with coefficient-one components of the ambient log formula. Its image is an ambient generalized center and hence meets the good open set. Over that set the restriction is snc and its nef data descend. The same reasoning applies to the ordinary companion. In particular, after effectivity is established, both restricted pairs have the appropriate dlt good-open property. Their zero centers agree: on a common carrier their log boundaries differ by the effective restriction of H, while every generalized zero center meets the open set where that restriction vanishes.

Here is the local effectivity and coefficient calculation, with the issue of nonfactorial intermediate strata made explicit. Suppose at some stage of the chain that U is the normal space already obtained, C is its effective generalized boundary, and S\subset\lfloor C\rfloor is the next prime. Carry along the ordinary dlt companion just constructed. Its floor has the same strata. Both adjoints are rational Cartier, although the individual trace M_U need not be rational Cartier on U. Fix a prime divisor on S^\nu whose coefficient is to be computed, and work at an analytically general point of its image, a codimension-two locus of U.

Take a general transverse surface slice there. This operation can be performed by fixing general values of local parameters along that locus on a simultaneous embedded resolution. We choose the parameters transverse to the finitely many resolution strata dominating the locus. At a general point in question the normal ambient space is Cohen–Macaulay: a normal space is S_2, and its non-Cohen–Macaulay locus has codimension at least three. The sliced surface is S_2 and regular in codimension one, hence normal. The transverse restrictions of the resolution formulas are crepant surface formulas, by complete-intersection adjunction. One can check this first away from the distinguished point and then extend the isomorphism of the rational Cartier adjoints. In particular, the ordinary surface pair is dlt along the coefficient-one branch. The boundary multiplicities computed on the slice are the multiplicities at the original general point.

We use the following local surface fact. An ordinary dlt surface germ along a coefficient-one branch is either snc for the reduced boundary, or is a plt germ with a smooth pair chart modulo a small cyclic group of some order r. The branch in the chart is a coordinate curve. The different of that branch alone is 1-1/r, and the local Cartier index of any integral divisorial sheaf divides r. This is the log surface classification of [31, Theorem 4.15(1),(3), pp. 119–120, and Proposition 4.18]. Its use here is in the analytic-germ category: the numerical plt resolution criterion gives a Hirzebruch–Jung string with the strict boundary meeting an end; the corresponding normal analytic surface germ is the cyclic quotient. It is a classification of the surface germ, and does not assert local factoriality of the higher-dimensional space U. In the snc case set r=1.

Write the additional effective boundary on the slice as a sum with coefficients d_j. Pulling to the cyclic chart and restricting to its smooth coordinate branch gives nonnegative integral intersection multiplicities k_j. Its contribution to the different is \sum_j k_jd_j/r. For the generalized part, restrict a common higher model and the line bundle representing p\bM to the higher surface. Define its trace on the sliced surface by this proper pushforward; the restricted higher log formula fixes the identification on the punctured surface. This avoids interchanging an arbitrary reflexive hull with specialization. The cyclic quotient fact makes rp times this trace Cartier. Consequently its pullback minus the higher nef bundle is an exceptional divisor with denominator dividing rp. It is effective by Lemma 2.3, because the higher bundle is nef on every curve of the projective surface resolution. It does not contain the strict coefficient-one branch, and its restriction to that branch has coefficient k/(rp) for some k\in\Z_{\geq0}. The resulting coefficient is therefore

This proves effectivity of the generalized different. Omitting the nef correction proves effectivity of the ordinary different carried along in the construction. The coefficient is at most one by the sub-lc log formula already established. The local analytic adjunction and nef-defect calculation also appears explicitly in the proof of [25, Lemma 3.1, pp. 5–6]; the local gdlt modification in that proof is not used here as a global modification theorem.

We can now repeat the construction. On a resolution the chosen chain is a chain of snc strata, and its successive restrictions have the two log formulas just considered. A zero stratum of any restricted formula is a zero stratum of the preceding one; its image is therefore an ambient center meeting the original good open set. This proves the center and gdlt assertions at every stage. Closures and normalizations introduce no finite cover of the chosen center: each selected component is generically isomorphic over it, so its normalization is its unique normalization. The restricted higher model is projective over that normalization and is compact Kähler. The higher line bundle remains nef, and its denominator still divides p, under restriction and subsequent pullback. This proves (11.1) and the asserted description of the b-data. A fixed chain on a common higher model also shows that these b-data are compatible under any birational comparison which preserves the general points of the chain.

It remains to prove the DCC assertion rather than infer a bound on all Cartier indices. If I is DCC, its positive elements, when present, have a positive minimum. The same is true after adjoining 1/p. A sum of such elements bounded by one consequently has a bounded number of nonzero summands. Finite sums of a nonnegative DCC set satisfy DCC: from any sequence of tuples of a fixed length one can successively choose a subsequence on which each coordinate is nondecreasing. Thus the numerator sums in (11.2) satisfy DCC. For bounded r, their images in that formula satisfy DCC. In a strictly decreasing sequence of displayed coefficients with first term c_0<1, the inequality

bounds r. A sequence starting at one reduces to this case after its first strict decrease. Therefore \mathcal D_p(I) is DCC, and induction proves the same for each finite iterate.□

Proof. The restricted morphisms factor through S by normality. They are projective, since restriction and this finite factorization preserve relative ampleness; they are bimeromorphic because the ambient step is an isomorphism at the general point of the stratum. Restriction of the relatively ample adjoint, and then pullback by the normalization, gives the asserted signs.

Take a common projective log resolution of the ambient step, carrying its common nef data and preserving the general points of the chosen snc chain. If a,b are its projections, Lemma 2.3 gives the actual divisor comparison

This difference is anti-nef over Z. Its support contains the full fibers over the non-isomorphism locus in Z: opposite relative ampleness makes the difference nontrivial over each such point, and the fiber-support assertion of Lemma 2.3 applies. The strict transform of the chosen stratum is not contained in \Supp F, since its general point lies in the common isomorphism open set.

Restrict the two log formulas along the common snc chain. At each restriction the residue canonical bundles are the same, and the common higher nef bundles are the same. They cancel. Each restricted difference is effective, because the next strict stratum is not contained in its support. At the last stage this gives (11.5) as an equality with its prescribed meromorphic identification. The coefficient of E in this difference is the increase of its generalized log discrepancy.

For strictness, first make E divisorial on a smooth higher stratum. This can be achieved by projective blowups of its successive centers; perform the same blowups in the ambient resolution and resolve further, preserving the generic chain. If the center of E is in the ambient exceptional locus, its entire inverse image in the common ambient model lies in \Supp F, by the full-fiber assertion. Therefore the center on the higher stratum lies in the support of the restricted effective Cartier divisor. Its defining local equation has positive order at E. This proves strictness on either side. In particular, the argument detects an intersection in arbitrary codimension on the original stratum; it does not require that intersection already to be a divisor there. The algebraic counterparts are [32, Lemma 2.8] and [36, Proposition 3.2]; the argument above uses the analytic projective negativity statement directly.□

Joint proof of Theorem 11.3, Proposition 11.4, and Theorem 11.5. For each d in increasing order we prove special termination, then the modification assertion, and, if d\leq3, gdlt termination. In dimension zero these statements are immediate. Fix d>0, assume all the appropriate assertions in smaller dimensions, and first prove special termination in dimension d.

Stabilizing centers. There are only finitely many divisorial steps, by Lemma 4.1; hence we may discard a finite prefix and consider only small steps. Generalized discrepancies do not decrease. A zero-discrepancy place on the positive side was consequently a zero-discrepancy place before the step, and strictness in Lemma 2.3 excludes its center from either exceptional locus. Thus every new center is the transform of an old center with unchanged general point. If the exceptional locus contains the general point of an old center, that center disappears. There are finitely many centers on the initial model, so after another finite prefix their list is unchanged and all steps are isomorphisms at their general points.

For each center choose a chain of coefficient-one strata at its general point. Transform this choice along the sequence. Lemma 11.1 gives the corresponding normalized strata T_i and their adjoints D_{T_i}. The b-line data on these strata are the same under their birational identifications, with one fixed higher denominator. The ambient boundary coefficients belong to a fixed finite set. The coefficients after adjunction therefore belong to one fixed DCC set, depending only on that set, the denominator, and the length of the chosen chain.

We now induct on the dimension e of a surviving center to show eventual disjointness. A point center is already disjoint after the stabilization just made. For a center of positive dimension e, its proper zero substrata have smaller dimension and there are finitely many of them. Discarding a further finite prefix, we may assume that both exceptional loci avoid every such substratum at every remaining step. In particular, the induced maps on T_i are isomorphisms in neighborhoods of their non-gklt loci.

Finitely many extractions on the stratum. Identify divisorial places over the T_i through common proper bimeromorphic analytic models. If a prime divisor E is extracted by T_i\dashrightarrow T_{i+1}, then E is a divisor on T_{i+1} and its center there is contained in the ambient exceptional locus. Otherwise the ambient isomorphism near its general point would provide its strict transform as a divisor on T_i. Lemma 11.2 gives

where the last inequality uses effectivity of the boundary. By monotonicity its discrepancy on T_0 is less than one. Its center on T_0 cannot be contained in \operatorname{Nklt}(T_0,A_{T_0}+\bM^T): the maps are isomorphisms near that locus and its transforms, so a valuation centered there cannot acquire a divisorial center through the transformations under consideration.

There are only finitely many such valuations by the off-non-gklt assertion of Lemma 4.2. In addition to its exceptional valuations, include the finitely many divisors already on T_0 having discrepancy less than one; these are components of its positive boundary. For any one of this finite set, the coefficients on the models where it is extracted form a strictly decreasing sequence: discrepancy never decreases, and each new extraction gives a strict increase. Those coefficients are in the fixed DCC set. Thus each valuation is extracted only finitely often. Altogether there are finitely many extractions.

Removing divisorial changes and stabilizing coefficients. After these extractions, the induced bimeromorphic maps extract no divisors. Lemma 4.1, applied to the span of prime (e-1)-cycles of the compact Kähler strata, shows that only finitely many further maps contract a divisor. In dimension one a proper bimeromorphic map between normal curves is already an isomorphism. We may therefore assume that every induced map is an isomorphism in codimension one.

With primes matched on this tail, their boundary coefficients do not increase, by Lemma 11.2. No component with coefficient zero can acquire a positive coefficient. Only the finite boundary support at the start of the tail needs to be considered, and DCC stabilizes each coefficient. Consequently

Let S_i be the normalization of their common image in the contraction base. Neither T_i\to S_i nor T_{i+1}\to S_i contracts a prime divisor. Indeed such a prime is matched in codimension one on the other stratum, but its center is in the corresponding ambient exceptional locus. Strict discrepancy increase would contradict coefficient stabilization.

We have thus obtained small diagrams over the varying S_i, with antiample and ample adjoints, respectively, and strict-transform boundary and b-data. The effective difference of Lemma 11.2 is now exceptional over both stratum models: its coefficient at a divisor on either model is zero by the matched data. If one of the induced birational maps is an isomorphism, the matched boundary and b-line identify its adjoints as actual rational bundles with their canonical meromorphic comparison. The restricted difference is then zero on a common model. The strictness assertion shows that this step has no ambient exceptional intersection with the stratum. Thus only nontrivial small stratum surgeries remain to be excluded.

Lifting the small surgeries in dimension e<d. The induction hypothesis in dimension e gives a projective crepant gdlt modification h_0:Y_0\to T_0 with globally strongly \Q-factorial source. Over S_0 run the relative elementary steps of Proposition 9.7 for its crepant adjoint. By gdlt termination in dimension e, this run is finite and ends at a model Y_1 whose adjoint is nef over S_0.

On a common smooth model of the initial lift, its endpoint, and T_1, write the actual comparisons as

Here P_0 pulls back the initial crepant adjoint, P_1 pulls back the adjoint of Y_1, and P_+ pulls back D_{T_1}. The relative steps give G\geq0 exceptional over Y_1. The restricted comparison just established, pulled through the initial crepant modification, gives F\geq0 exceptional over T_1. Lemma 10.5 applies: P_1 is nef over S_0 and D_{T_1} is ample there. It gives P_1=P_+ and a projective crepant morphism h_1:Y_1\to T_1.

This endpoint is again a gdlt modification of the required type. The elementary steps preserve gdlt and effectivity. They extract no prime divisor, and the stratum map below is small. Any prime on Y_1 exceptional over T_1 was therefore exceptional on Y_0 over T_0. If it survives, its coefficient is still one, since boundaries are pushed forward. Crepancy identifies that coefficient with its zero-discrepancy extraction over T_1.

Repeat over S_1,S_2,\ldots. Each nontrivial stratum surgery forces at least one lifted elementary step. Indeed its negative morphism has a contracted curve C with D_{T_i}\cdot C<0. Projectivity gives a curve on the lift dominating C with positive degree, on which the pulled-back adjoint is negative. The lifted model is therefore not already nef over S_i. Infinitely many nontrivial stratum surgeries would give an infinite concatenation of elementary gdlt steps in dimension e. This contradicts the lower-dimensional gdlt termination hypothesis, which allows varying compact contraction bases. There are only finitely many such surgeries. The preceding isomorphism case now proves disjointness for the chosen center.

This completes the induction on e. Taking the maximum of the finitely many resulting indices gives special termination in dimension d, on both sides of every remaining step. This proves Theorem 11.3 in that dimension.

Constructing the modification in dimension d. Now let (V,A+\bM) be an arbitrary rational glc pair as in Proposition 11.4 in dimension d. Take a projective log resolution h:W\to V with W smooth, carrying the higher nef bundle. Let

The pair (W,\Gamma+\bM) is gdlt: its boundary is snc and its data descend to W. Its adjoint has the actual comparison

The support of Q is contained in \lfloor\Gamma\rfloor. Run relative elementary steps over V using Proposition 9.7. On every model the pushforward of (11.6) remains the identity of its adjoint with the pulled-back original adjoint plus the effective exceptional transform of Q. Every surviving component of that transform is a floor component.

If this run were infinite, special termination in dimension d, just proved, would make its tail disjoint from the floor. A contracted curve on such a tail lies outside the support of Q and has nonnegative degree against this effective rational Cartier divisor. Its degree against the pulled-back original adjoint is zero. This contradicts negativity of the elementary step. Thus the run is finite; continuation ensures it ends at relative nefness. On the endpoint Y, the remaining transform Q_Y is nef over V and exceptional, so Lemma 2.3 gives Q_Y\leq0. It is also effective, hence zero. The comparison is crepant. The resulting model is projective over V, compact Kähler, globally strongly \Q-factorial, and gdlt, by the relative construction. Any prime exceptional over V is the transform of an exceptional prime on W, and has coefficient one in the pushed boundary. This proves Proposition 11.4 in dimension d.

Gdlt termination in dimension d\leq3. Finally suppose d\leq3 and consider any infinite elementary gdlt sequence in this dimension. Discard its finite divisorial prefix and apply special termination. On the resulting tail all exceptional loci avoid \lfloor A_i\rfloor. Replace its boundary by A_i-\lfloor A_i\rfloor on each model, retaining the same b-line. The resulting pairs are effective and gklt. To check the latter, discrepancies cannot decrease on removing an effective rational Cartier floor. Each old zero-discrepancy place has center in the floor, and the order of its pullback is positive, so its discrepancy becomes positive. Equivalently this is the snc calculation on a fixed carrier and the gdlt good-open condition.

The floor has degree zero on every contracted curve on both sides, because it is disjoint from the exceptional loci. Thus all steps of this tail are still negative and positive, respectively, for the new adjoints. Their boundaries are strict transforms, their nef data are unchanged, and their one-ray condition for global line-bundle degrees is unchanged. They are consequently elementary gklt steps in the category of Theorem 10.6. That theorem excludes an infinite tail. This proves Theorem 11.5 in dimension d and completes the triangular induction through dimension four.□

Proof. Choose a projective log resolution p:W\to V carrying \bM and resolving all the divisor comparisons below. Write

These are the comparisons of actual rational sheaves from Lemma 9.3. Strong \Q-factoriality makes N and M_V rational Cartier, and

by Lemma 2.3. After increasing W, the support of C_W+H is simple normal crossing. The crepant higher boundary at parameter t is C_W+tH.

The divisor H is nonzero. Otherwise p^*T=M_W is nef. Since p is projective, there is a compact curve \widetilde C\subset W mapping onto C with positive degree: take a component dominating C in p^{-1}(C) and intersect it with sufficiently many general relative hyperplanes. The projection formula would give

Here only degrees on curves are used; no absolute curve criterion for analytic nefness is needed.

For the finitely many prime components E with h_E=\operatorname{coeff}_E H>0, put c_E=\operatorname{coeff}_E C_W. The log-smooth discrepancy criterion gives

Indeed all coefficients of C_W+aH are strictly below one, and components with h_E=0 impose no new bound. Formula (12.3) proves finiteness, rationality, attainment, and s>a, as well as the claimed singularities at and below s.□

Proof. Take h_1 by Proposition 11.4. Suppose the current h_i has been constructed. We choose a nearby klt parameter, run a finite relative program, and identify its endpoint with a crepant model of X_{i+1}.

Choosing a nearby klt parameter. For t\in[a,s], define its crepant boundary by

This boundary depends affinely on t. Its nonexceptional coefficients are b+tn with b,n\geq0. Its exceptional coefficients equal one at s. There are only finitely many exceptional primes, so for a rational t_i<s sufficiently close to s, with t_i\geq a, the boundary \Delta_i(t_i) is effective. In particular, a nonexceptional coefficient zero at the positive parameter s has b=n=0 and stays zero. The pair downstairs is generalized klt for a\leq t<s: its discrepancies are affine combinations of the positive discrepancies at a and the nonnegative ones at s. Thus the crepant pair at t_i upstairs is effective and generalized klt.

Choose a curve in the ray of f_i. By the one-ray condition on global rational line-bundle degrees, there is a positive rational number q_i such that A_i(t_i)-q_iA_i(s) has zero degree on all f_i-contracted curves. Apply Lemma 3.1, using the generalized klt antiample adjoint at t_i, to obtain a rational line bundle L_i on Z_i with

After pullback by h_i, this is an actual identity modulo a rational bundle from Z_i, not merely an equality of numerical classes.

Running the finite relative program. The composite Y_i\to Z_i is projective and bimeromorphic. Apply Proposition 9.9 to the effective generalized klt pair at t_i to obtain some finite relative elementary run ending nef over Z_i. Transforming (12.4) through each step shows that the same run is negative, and positive after each flip, for the full adjoint at s. It is therefore a generalized dlt run at s, by Proposition 9.7, and its endpoint is nef for this adjoint as well.

Identifying the endpoint. To check the endpoint, on a common resolution write

Here P_0 is the pullback of h_i^*A_i(s), P_j is the pullback of the lifted endpoint adjoint, and P_+ is the pullback of A_{i+1}(s). Negativity for the lifted run gives G_j\geq0, exceptional over its endpoint, and negativity for the original small step gives F\geq0, exceptional over X_{i+1}. Both endpoint adjoints are nef over Z_i, and A_{i+1}(s) is ample there. Lemma 10.5 applies. Explicitly, if p and q denote the maps to these two targets, then P_j-P_+=F-G_j is q-nef with pushforward -q_*G_j\leq0; its negative is p-nef with pushforward -p_*F\leq0. The two negativity inequalities give P_j=P_+. Relative ampleness and projective connected fibers then give the crepant projective morphism from the lifted endpoint to X_{i+1}.

Every prime exceptional for this morphism is the transform of an h_i-exceptional prime: the original step is small and the lifted run extracts none. If it survives, its coefficient one is preserved by the boundary pushforward rule. This proves the required assertion for h_{i+1}. The string is nonempty, since h_i^*A_i(s) has negative degree on a curve lifting an f_i-contracted curve and so is not nef over Z_i. Induction proves the lemma. The choice of t_i is made separately at each finite stage.□

Proof of Theorem 12.1. From an infinite sequence we will construct successively larger thresholds with fixed coefficient data. At each repetition, special termination will let us remove the floor while retaining an infinite sequence with the required signs.

Step 1: the infinite tail and its first threshold. Suppose there is an infinite sequence. By Lemma 4.1, it has only finitely many divisorial steps. Truncate after the last of them and relabel the resulting sequence by X_1\dashrightarrow X_2 \dashrightarrow\cdots. Taking actual reflexive traces transports (12.1) through this finite prefix. The effective divisor remains effective and rational, and a projective common higher model carries the pullback of the original nef line bundle. Thus the relabelled input still has the required expression.

We will repeat the following construction. Its input is an infinite small elementary sequence with rational families

and a rational a\geq0 such that the pairs at a are generalized klt, and both T_i and A_i(t) have the negative-to-positive signs of each step for all rational t\geq a. The divisors B_i,N_i are effective strict transforms. Initially a=0, the nef part at zero is zero, and

so these conditions hold.

Let s be the threshold of the first pair in (12.6). Lemma 12.2 applies to a contracted curve and gives s\in\Q, a<s<\infty. Since A_i(s) is negative on the input side and ample on the positive side at each step, discrepancy monotonicity shows that every pair in the sequence at s is generalized lc. Lemma 12.3 gives an infinite concatenation of generalized dlt elementary steps at s, with compatible crepant endpoint morphisms to the X_i.

Step 2: deleting the floor after special termination. Apply Lemma 4.1 and Theorem 11.3 to this concatenation. It has a tail consisting of small steps whose exceptional loci, on both sides, avoid the support of the coefficient-one boundary. Every lifted string is finite and nonempty, so an endpoint occurs beyond any prescribed finite prefix. Start the tail at such an endpoint h:Y\to X_i and relabel. Write the union of the supports of B_i,N_i as \bigcup P, with coefficients b_P,n_P\geq0. If \widehat P denotes the strict transform on Y, the crepant boundary at s is

Set

Then

Deleting the floor of a generalized dlt pair gives a generalized klt pair with the same nef data. Indeed every place of discrepancy zero has center in that floor, whose rational Cartier pullback has positive order at the place; all other discrepancies were already positive. Consequently (Y,\widetilde B+s\widetilde N+s\bM) is generalized klt. Both divisors in (12.7) are effective.

Step 3: retaining the signs and increasing the threshold. Let their strict transforms define the new family on every model of the lifted tail. Its signs require checking for the testing direction, as well as for its adjoint. Do this over each original flip base Z_i. At the initial endpoint of its lifted string, the trace comparison gives

To see the support assertion, the difference between pullback and strict transform of N_i, and the difference between pullback and b-line trace of M_{X_i}, are h_i-exceptional. Every such prime has coefficient one in \Delta_i(s). The remaining difference consists of the discarded n_P\widehat P, also supported in the floor. This is a comparison of actual rational sheaves; a sign for F_i is unnecessary.

On X_i, both A_i(s) and T_i have negative degree on the contracted ray, and their ratio is a positive rational number c_i. Lemma 3.1, applied using any nearby generalized klt parameter below s, gives a rational line bundle Q_i on Z_i such that

Transform this identity through the finite lifted string. The tail is small, so all its boundary divisors, traces, and F_i transform strictly. The support of the transform of F_i remains in the transformed floor. All contracted and flipped curves are disjoint from that floor. Equation (12.9) therefore shows that \widetilde T=\widetilde N+M_{\mathrm{trace}} has negative degree on every contracted ray and positive relatively ample transform on the other side. For the ampleness assertion, the same identity, with the floor bundles trivial near the exceptional fibers, identifies its restriction there with a positive rational multiple of the full ample adjoint; fiberwise ampleness is the relative ampleness criterion.

Likewise subtracting the floor changes neither sign of the adjoint at s. On each side, for every rational u\geq s, the new adjoint is

so it has the same negative-to-positive signs. At later endpoints the retained coefficients still obey (12.7): the downstairs maps are small, and an exceptional divisor of an endpoint morphism cannot become the transform of a downstairs prime. All boundaries upstairs are pushforwards, hence strict transforms on this tail. Thus the families constructed over consecutive bases agree.

We have obtained another infinite small elementary sequence satisfying the same conditions, now with a=s. The generalized klt condition at s holds throughout the tail by discrepancy monotonicity. It also implies the generalized klt condition at parameter zero, by (12.2). Lemma 12.2 shows that its next threshold is finite, rational, and strictly larger than s. The construction can therefore be repeated indefinitely if the original sequence was infinite. It produces

Step 4: the fixed data for analytic ACC. Let I be the finite set consisting of zero and the coefficients of the first B, and let J consist of zero, the coefficients of the first N, and 1/p, where p>0 makes pM_{W_0} integral on the original nef carrier W_0. At every repetition, the new boundary and testing coefficients are obtained solely by omissions from the previous ones, as in (12.7). Newly extracted coefficient-one primes are removed before the next threshold input. The boundary coefficients therefore stay in I, the testing-boundary coefficients stay in J, and the nef part at parameter zero remains zero. On every required higher model the testing nef data are \frac1p times the pullback of the same integral nef line bundle. The denominator p is fixed. Common projective models are needed only for finitely many maps at a time and do not change p.

We verify explicitly that each s_j belongs to the analytic generalized threshold set with these data. Choose a general point of a computing center on its compact input, and a relatively compact Stein neighborhood U of that point. Restrict a projective log resolution carrying the data to U, and denote it by g:W\to U. The computing place still meets this restriction. Hence the local threshold equals s_j: the pair is lc at s_j, whereas the same place has negative discrepancy for every larger parameter.

Let \cL represent the integral line bundle pM_W. The coherent sheaf g_*\cL has rank one on the isomorphism locus of g. Cartan’s Theorem A supplies a section nonzero at a point of that locus, and thus a nonzero section of \cL on W. Its divisor P' is Cartier and represents \cL. In particular P' is g-nef and M_W is represented by \frac1pP'. The induced meromorphic identification gives precisely the original trace and pullback defect. Compatible local canonical representatives may be chosen in the same way from the coherent canonical sheaf. Neither choice changes the boundary or testing-boundary coefficients, the discrepancies, or the factor 1/p. The testing boundary is rational Cartier because its components are restrictions of global rational Cartier divisors on the strongly \Q-factorial input.

The analytic generalized threshold theorem [25, Theorem 1.1 and Theorem 3.3; see also Definition 2.3 and Remark 2.5] consequently applies in dimension four with the fixed DCC sets I,J. Its nef summands are actual relatively nef Cartier divisors with the fixed weight 1/p; it requires no common bound for Cartier indices of their downstairs traces. It prohibits the strictly increasing sequence (12.10). This contradiction proves the theorem.□

Proof. Choose a projective log resolution p:W\to X as in the statement. We first describe how to transfer a nef adjoint on a minimal model reached by a finite program from W to a rational b-line summand over X.

Suppose that a finite program on W ends at a model Y, and that H_W and H_Y are its initial and final actual rational adjoints. Both morphisms defining each step are projective. The main components of the successive fiber products of their graphs, followed by normalization and resolution, therefore give a diagram

whose two arrows are projective modifications. The construction uses only the finitely many steps of this program: projectivity follows successively by base change and composition. In particular r=pa is projective over the original X, and U is compact Kähler.

The stepwise comparisons in Lemma 2.3, pulled back to this common model, give

Set M_U=q^*H_Y. When H_Y is nef, so is M_U: pull back metrics with arbitrarily small negative curvature and compare the pulled-back fixed form with a Kähler form on U. In the projective case U is itself projective, and the pullback of a nef rational bundle is analytically nef by the projective metric characterization. For a common multiple m, the trace of M_U is represented by the rank-one reflexive sheaf

Coherence follows from properness, and global strong \Q-factoriality on X makes a further reflexive power invertible. By Lemma 9.3, tracing (13.2) gives

Indeed this is the usual tensor identity at each codimension-one point of X, where r is an isomorphism, and both sides extend reflexively. It retains the actual identifications of the line bundles. In particular, no global meromorphic frame of either canonical bundle or of M_U is chosen.

Non-uniruled resolution. By [39, Theorem 1.1], K_W is analytically pseudo-effective. Starting with the smooth empty-boundary pair (W,0), apply Lemma 3.2 and Proposition 3.4 whenever the current canonical bundle is not nef. Lemma 7.1 preserves its pseudo-effectivity, and Lemma 7.2 excludes a fiber-type contraction at every stage. Terminality persists by Lemma 2.3. Corollary 4.7 rules out an infinite sequence. Continuation from every non-nef model thus gives a finite program ending at a terminal compact Kähler Y with analytically nef actual rational canonical bundle K_Y.

Apply (13.2) with H_W=K_W and H_Y=K_Y. The reflexive trace of a^*K_W on X is K_X: over the common codimension-one open set the canonical sheaves are canonically identified, and normality determines their reflexive extensions. Equation (13.3) therefore yields

Take N=r_*G+B. This argument uses the trace of K_W; it does not require the discrepancies comparing K_W with p^*D to be nonnegative. The terminal fourfold result used here was established independently of Theorem 12.1.

Uniruled Moishezon resolution. The smooth Kähler manifold W is then projective. By Lemma 8.1, choose an effective rational SNC klt boundary \Gamma satisfying

This adjoint is analytically pseudo-effective, hence numerically pseudo-effective in the projective sense by [2, Proposition 1.2].

Run the ordinary projective klt MMP on (W,\Gamma). Divisorial steps are finite, and every remaining flip sequence terminates by [3, Theorem 1.1], applied to a pseudo-effective NQC lc fourfold with zero nef part. These are projective varieties over a point, as required by that theorem’s conventions. Pseudo-effectivity persists by Lemma 7.1, so Lemma 7.2 excludes a fiber-type negative contraction. The resulting model Y has nef rational adjoint H_Y=K_Y+\Gamma_Y.

Form (13.2) for this finite program. Since E_0 is exceptional over X, the reflexive trace of a^*H_W in (13.4) is D. Thus (13.3) gives

Take N=r_*G. The projective fourfold theorem has supplied a nef adjoint; fourfold abundance is not used.□

Proof. Construction. Let b:B\to\Pbb^4 be the blowup of a point, let H=b^*\cO_{\Pbb^4}(1), and let F be the exceptional divisor. The graph of projection from that point embeds B in \Pbb^4\times\Pbb^3, and the restriction of \cO(1,1) is 2H-F. Thus 2H-F is very ample. Since H is globally generated,

is very ample as well. Take a general smooth member U\in|6H-2F|. Its image U_0\subset\Pbb^4 is smooth away from the blown-up point. The quadratic jets of the defining sextics at that point are arbitrary, so a general member has a nondegenerate quadratic initial term. The holomorphic Morse lemma identifies its unique singularity with

The restriction \pi:U\to U_0 resolves this node, with exceptional quadric

Adjunction gives

since K_B=-5H+3F.

Weak Lefschetz for the smooth ample threefold U\subset B gives

This is the blowup calculation underlying the classical example in [40, Definition 5.1, Remark (b)]. Let C be a smooth elliptic curve and set X=U\times C. Its canonical divisor is the pullback of (A.1), so is effective. If \ell is either ruling in E\times\{c\}, then K_X\cdot\ell=-1; in particular K_X is not nef.

All negative contractions have the same target. Put D=E\times C and let h=\{e\}\times C be a horizontal curve. A curve not contained in D has nonnegative degree against both the effective divisor D and the nef hyperplane pullback. Thus every K_X-negative curve lies in D. The two rulings of E have the same class in H_2(U,\R): both have degrees 0,-1 against the displayed basis of H^2(U,\R). Künneth and H^1(U,\R)=0 now show that every integral curve \gamma\subset D has class

Here a is the sum of the projection degrees to the two factors of E=\Pbb^1\times\Pbb^1, and b is the projection degree to C. Moreover K_X\cdot\gamma=-a, so negativity means a>0.

Let f:X\to Z be any nontrivial projective birational contraction with connected fibres, normal compact Kähler target, and -K_X relatively ample. Choose a Kähler class \omega_Z and set \alpha=f^*\omega_Z. For a curve, vanishing of its degree against \alpha is equivalent to being contracted by f. In particular \alpha\cdot\ell\geq0 and \alpha\cdot h>0: the horizontal curve cannot be contracted because K_X\cdot h=0. A contracted curve \gamma has the form (A.2) with a>0, and

It follows that b=0 and \alpha\cdot\ell=0. Thus all rulings in every E\times\{c\} are contracted, and f is not small.

Set q=\pi\times\id_C:X\to U_0\times C. Its nontrivial fibres are the quadrics E\times\{c\}, which are connected by their rulings, so f is constant on every q-fibre. Conversely, every curve in an f-fibre has the form (A.2) with b=0, and is contracted by q. A connected projective fibre is connected by chains of curves, so q is constant on every f-fibre. The two targets are isomorphic: the reduced image of (q,f) in their product projects properly, bijectively and bimeromorphically to each normal target, hence finitely and isomorphically. Therefore q is the only possible first negative birational contraction. It is divisorial, and its relative numerical rank is one because both quadric rulings have the same nonzero numerical class.

Failure of analytic local \Q-factoriality. Near the singular locus of U_0\times C, consider the prime analytic divisor

On the blowup of the singular axis, its strict transform meets each exceptional quadric in a ruling line. If nP were Cartier near an axis point for some integer n>0, its pullback would have the form n\widetilde P+kE_{\mathrm{exc}}. The associated line bundle restricts trivially to the quadric over that point, because it is pulled back from a line bundle at a point. On the other hand its restriction is

up to interchanging the factors. Triviality forces k=0 and n=0, a contradiction. The target is not analytically locally \Q-factorial. Since K_X is not nef and there is no small negative contraction, there is no permissible first step under that local convention.

Compatibility with the global convention. The same example has a valid one-step global program. Indeed, -E is \pi-ample, so -K_X is q-ample, and adjunction on U_0 gives

which is nef. On the blowup of the node the exceptional divisor has log discrepancy two; its smooth product with C is a log resolution with the same log discrepancy. Hence the target is terminal, in particular klt.

It remains to check that the local divisor P is not a global factoriality obstruction. The displayed integral cohomology of U and Hodge decomposition give H^1(U,\cO_U)=H^2(U,\cO_U)=0 and \Pic(U)=\Z[H|_U]\oplus\Z[E]. The exponential sequence and Künneth therefore give

Strict transform and divisor pushforward identify

Every surviving class descends to a Cartier class, either from the hyperplane bundle on U_0 or from a line bundle on C. Chow’s theorem makes every global analytic Weil divisor on this projective target algebraic, so every such divisor is Cartier. GAGA identifies its global coherent rank-one reflexive sheaves with algebraic divisorial sheaves, which are therefore invertible as well. The plane P is defined only on an analytic neighbourhood; it need not extend to a divisor on the whole compact space.□