Termination of generalized log canonical flips on compact Kähler fourfolds

Proof. For a diagram T\to Z\leftarrow T^+, take the reduced component of T\times_ZT^+ containing the graph over the common isomorphism open. Its projections are projective, and finite normalization preserves projectivity. Adjoin the remaining spaces and the carrier one at a time by the same construction. Resolving the resulting dominating component gives a common smooth model. Analytic resolution and principalization allow the stated simultaneous choices by projective modifications; see [2, Theorems 1.6, 1.10 and 13.2(1)]. Base change, restriction to a closed analytic subspace, finite normalization, and composition preserve projectivity. For composition in the present compact setting, one can choose a uniform sufficiently large twist of a relative ample bundle by the pullback of the next one. This also proves compatibility with any finite chain of the small diagrams and divisorial contractions considered below.

A smooth space projective over a compact Kähler space is compact Kähler: a relative ample bundle supplies positivity in the fiber directions, and adding a sufficiently large multiple of a pulled-back Kähler form supplies positivity in all directions. Finally, choose Kähler classes on X' converging to c_1(M'). Their pullbacks to W have semipositive representatives. Adding positive multiples tending to zero of a fixed Kähler class on W gives Kähler classes converging to c_1(M_W). Thus M_W is analytically nef.□

Proof. The trace definition gives r_*J_{W/T}=0. On an r-contracted curve, -J_{W/T}=M_W-r^*M_T has the same degree as M_W, hence has nonnegative degree by projection formula from the nef carrier. Analytic negativity gives J_{W/T}\geq0. The final identity follows from functoriality of pullback and the fact that the b-divisor descends on W.□

Proof. Choose a smooth log resolution r:W\to T, projective over T and carrying \M, with divisorial exceptional locus and simple normal crossing support for all divisors in the crepant equation

Such a simultaneous resolution is available by Lemma 3.1. The assumed inequalities give coefficients at most one, respectively strictly below one, for every global prime of \Delta_W. The same bounds hold on their local branches. On any further local model the nef part is pulled back from W, so its generalized discrepancies are the ordinary discrepancies of the simple normal crossing subpair (W,\Delta_W). The simple normal crossing discrepancy criterion proves the asserted local generalized inequalities, including when some coefficients of \Delta_W are negative.

Put J=J_{W/T}. The ordinary crepant boundary for (T,0) is

Both r^*B and J are effective. After resolving their joint support as above, every coefficient of \Delta_W^{(0)} is therefore strictly below one in the positive case. The same local simple normal crossing criterion gives ordinary klt singularities.

For the terminal assertion, the relative canonical divisor is exceptional, and at an exceptional prime R of W its coefficient is

These positive coefficients persist on every local branch. A local place exceptional over the smooth space W has ordinary log discrepancy at least two over W: the Jacobian determinant of a smooth model carrying it has positive integral order along an exceptional divisor. Its discrepancy over T adds the nonnegative order of K_W-r^*K_T, and hence remains greater than one. Local exceptional places already represented by divisors on W satisfy the displayed strict inequality. Thus T is locally ordinary terminal. For a normal complex fourfold with a rational canonical line bundle, local ordinary terminality implies smoothness in codimension two by [21, Lemma 2.3], giving the last assertion.□

Proof. We first recall the fiber description for a proper bimeromorphic morphism v:V\to S of normal spaces. Its positive-dimensional-fiber locus is closed analytic. Over its complement the map is finite locally on S, hence an isomorphism by normality. Stein factorization shows that every fiber is connected, because its finite bimeromorphic factor over the normal space S is an isomorphism. A reduced compact connected positive-dimensional fiber has no zero-dimensional irreducible component: such a component would be a singleton disjoint from the finitely many other closed components. Hence no point in such a fiber is a local isomorphism point. The exceptional locus of v is exactly the full inverse image of its positive-dimensional-fiber locus.

Let U\subset Z be the open set over which f is an isomorphism. On U, D_T defines a \Q-Cartier divisor D_U. The actual Weil coefficients of the two adjoints agree under the identification of prime divisors. Since f^+ is small, it follows that D_{T^+}|_{(f^+)^{-1}U}=(f^+)^*D_U. A positive-dimensional projective fiber over U would contain a curve on which this divisor has degree zero, contrary to f^+-ampleness. Thus f^+ is an isomorphism over U. Interchanging the two sides and using f-ampleness of -D_T gives the reverse inclusion of the isomorphism opens. This proves the equality of exceptional images and (3.4). Smallness gives \dim L,\dim L^+\leq2. Over a general point of each component of A the fiber has dimension at least one, so the fiber-dimension theorem gives \dim A\leq1.

The graph projections are projective and surjective, and the map from G to its component in T\times_ZT^+ is finite. A prime divisor of either side maps to a prime divisor of Z by smallness. On the graph it therefore corresponds to the same prime place on the other side, with the same adjoint coefficient. Consequently p_*F=q_*F=0. On a p-contracted curve, -F has the degree of q^*D_{T^+}, which is nonnegative. Analytic negativity gives F\geq0.

For a curve C contracted by h, the projection formula and the two ample signs give

Here an image that is a point contributes zero. At least one image is a curve, because the finite map to the fiber product cannot contract C to a point. Thus at least one summand is strictly positive.

The support of F is contained in h^{-1}(A), since both projections identify the common isomorphism open and the adjoints agree there. Conversely, take x\in h^{-1}(A) and write z=h(x). The map h:G\to Z is proper bimeromorphic, so its fiber over z is connected. It is projective and positive-dimensional: its projection to the positive-dimensional fiber f^{-1}(z) is surjective. The fiber description above places x on a positive-dimensional irreducible component. Successive projective hyperplane cuts through x give an irreducible curve C in that fiber through x. If x were outside \Supp F, then C would not be contained in the effective \Q-Cartier divisor F, so F\cdot C\geq0. The preceding strict inequality excludes this. Hence (3.7) holds.

Follow a global place E on a common proper model dominating G and the nef carrier. Subtracting its two crepant equations cancels the same trace of \M and gives (3.8). The order of an effective \Q-Cartier divisor is positive precisely when the center of the place is contained in its support. Proper images of centers give

Together with (3.4) and (3.7), this proves both center equivalence and the stated exact strictness criterion. If the centers are not contained in these loci, their general points correspond on the common open and the order is zero. Finally, a place centered at a divisor on a normal model is that divisorial place. Smallness identifies these prime divisors on the two sides, proving equivalence of exceptionality.

Since both morphisms are nonisomorphisms, A is nonempty, so F has a prime divisor R in its support. The graph is a fourfold, hence \dim R=3. Its finite map to the graph component has image in L\times_ZL^+, whence

This is the remaining inequality in (3.5).□

Proof. For a divisorial step put E_j=D_j-v_j^*D_{j+1}. It is exceptional over Y_{j+1} and has strictly negative degree on every v_j-contracted curve. Negativity gives E_j\geq0. Every point of an exceptional fiber lies on a curve in that fiber, by the connected projective fiber argument in the proof of Lemma 3.4. The negative degree therefore forces that point into \Supp E_j. In particular the coefficient of E_j at each contracted prime is positive. Subtracting crepant equations gives nondecreasing discrepancies and this strict increase. For a small step the same assertions, with an effective correction exceptional over its output, follow from Lemma 3.4.

Pull the corrections of all steps to W and add them. This gives the effective divisor in (3.9). Each prime in a summand is exceptional over that step’s output, and remains exceptional over every later output. Indeed, if that place became a prime divisor on a later normal model, that prime would have a strict-transform prime on each earlier model, since none of the intervening steps extracts divisors. This contradicts its earlier exceptionality. Thus the sum is exceptional over Y_m. Its nonnegative coefficients also show that a strict increase from a contracted prime cannot be canceled later. For m=0 the correction is zero and the assertions are immediate.□

Proof. Write r:W\to Y and t:W\to T_+, and put H=P-P_+=E_+-E_Y. Then r_*H=r_*E_+\geq0. On an r-contracted curve, -H has the degree of P_+, which is nonnegative because the curve lies over a point of S. Negativity gives H\geq0. Likewise t_*(-H)=t_*E_Y\geq0, while H is t-nef because P comes from an S-nef divisor. Negativity applied to -H gives H\leq0. Thus P=P_+.

For a curve C in an r-fiber, equality gives t^*D_+\cdot C=P\cdot C=0. The image t(C) lies in an S-fiber, where D_+ is ample, so t(C) is a point. Every r-fiber is projective and connected. If an irreducible component of one such fiber had positive-dimensional image under t, a curve in that projective image and projective hyperplane cuts of its inverse image would supply a curve on which t is nonconstant. Thus each component maps to a point, and connectedness makes those points equal. The map t is constant on every r-fiber.

The reduced image of (r,t):W\to Y\times_ST_+ is a proper analytic graph with one-point fibers over Y. Its projection to Y is therefore finite and bimeromorphic, hence an isomorphism by normality of Y. This gives the morphism \varphi. Its graph is a closed subspace of Y\times_ST_+, whose projection to T_+ is projective by base change from Y\to S; hence \varphi is projective. Finally, pushing the equality P=P_+ to Y gives D_Y=\varphi^*D_+ as actual divisors.□

Proof. This is the rational-coefficient version of [19, Lemma 4.1]; we include its localization argument. The Borel–Moore localization sequence for B\subset Y contains

The outside groups vanish by \dim B<k and the real dimension bound for an analytic triangulation. Thus restriction from Y is an isomorphism in degree 2k. Compose restriction from X with its inverse. An irreducible compact analytic k-cycle meeting U is sent to its strict transform on Y; analyticity of that transform follows by proper mapping through the given diagram. A cycle contained in A is sent to zero. Conversely, no k-cycle of Y is contained in B, and its strict transform on X maps to its class. This proves the surjection on cycle spans. If A contains a k-dimensional irreducible analytic subspace, its nonzero class is in the kernel, giving strict inequality.

For a transformation that extracts no prime divisors, choose the common isomorphism open with target complement of codimension at least two. Every contracted source prime lies in the source complement. The preceding result with k=n-1 gives a strict drop at each such contraction. The ranks are nonnegative integers, so there can be only finitely many drops. For a small transformation both complements have codimension at least two, and applying the result in both directions gives equality. Finally, (3.4) gives the common open for a small fourfold diagram. Applying the first assertion with k=2, A=L and B=L^+ proves the last claim.□

Proof. By Lemma 3.4, discrepancies do not decrease, and a place is exceptional over X_i exactly when it is exceptional over X_{i+1}. Consequently the sets

form a decreasing sequence. Proposition 2.2 makes the first of them finite and gives a positive lower bound for all initial discrepancies. Monotonicity preserves that bound. A decreasing sequence of subsets of a finite set stabilizes, which gives (4.1). We reindex this tail.

If \mathcal I is empty, take the identity as the first junction. Otherwise apply Proposition 2.3 at the first index to extract exactly \mathcal I. Its hypotheses hold: the model is strong, the pair is generalized klt with rational effective boundary, and the fixed nef data are carried on a higher model projective over X_i. The resulting morphism is projective bimeromorphic and has a strong normal irreducible compact Kähler source. Crepancy gives the coefficient 1-a_i(E)\in[0,1-\varepsilon] on an extracted prime; all other boundary components are strict transforms of B_i. Thus the crepant boundary is effective. Notice that a place with a_i(E)=1 is extracted with coefficient zero.

Any place exceptional over this source is exceptional over X_i: a prime on X_i has a strict transform on every proper birational model. Such a place is not one of the extracted primes, so its discrepancy is strictly greater than 1 by (4.1). Crepancy and the positive lower bound prove generalized terminality. This constructs the first junction.

Suppose the junction over X_i has been constructed. The coefficient of S_{i,E} changes in (4.3) from 1-a_i(E) to 1-a_{i+1}(E). Both are nonnegative because E belongs to the stabilized set at both indices, and the latter is no larger than the former. The divisor subtracted from \Theta_i is effective and \Q-Cartier on the strong model Y_i. Its pullback to a higher model is effective; hence subtracting it only increases discrepancies. In particular, \Theta_i' remains effective and (Y_i,\Theta_i'+\M) remains generalized terminal. Set

We first compare this adjoint with the positive model downstairs. By Lemma 3.1, choose a smooth common model W, carrying \M, with projective maps q:W\to Y_i and r:W\to X_{i+1}, all over Z_i, and put

The coefficient change makes q_*F=0. Indeed, at a prime S_{i,E} with E\in\mathcal I, the coefficient of the crepant boundary computed from X_{i+1} is 1-a_{i+1}(E), the new coefficient in \Theta_i'. Every other prime on Y_i corresponds, by smallness downstairs, to a prime on X_{i+1} and has its transported boundary coefficient. The canonical representatives and the trace of the same b-divisor agree in this calculation. On a curve contracted by q, the divisor -F has nonnegative degree: q^*D_i' has degree zero and r^*D_{i+1} is pulled back from the f_i^+-ample divisor D_{i+1}. The negativity lemma therefore gives F\geq0. Moreover every prime on X_{i+1} is represented on Y_i, where the coefficient of F is zero. Thus F is exceptional over X_{i+1}.

Use the finite generalized klt part of Proposition 2.4 for D_i' over Z_i. The map Y_i\to Z_i is projective bimeromorphic to a normal compact Kähler base, the source is strong and generalized klt, and a common higher model projective over Y_i carries the fixed data with the required relative nefness. The proposition supplies a finite elementary program ending at a model \overline Y whose adjoint \overline D is nef over Z_i. On a further common model, write again q for the map to Y_i and \bar q for the map to \overline Y. Lemma 3.5 gives

Together with (4.4), these are the two effective comparisons required by Lemma 3.6. That lemma gives a projective morphism h_{i+1}:\overline Y\to X_{i+1} and the actual crepant identity \overline D=h_{i+1}^*D_{i+1}.

This program cannot have lost a marked prime S_{i,E}. Its discrepancy immediately after the coefficient change is a_{i+1}(E); by the crepant identity its discrepancy at the end is again a_{i+1}(E). If a divisorial step had contracted it, the strict comparison for that step would have increased this discrepancy, and subsequent comparisons could not decrease it. Nor can the program have lost any other prime on Y_i. Such a prime corresponds to a prime on X_{i+1}, and that prime must have a strict transform on \overline Y. An elementary program extracts no prime, so a prime once contracted cannot reappear. We have excluded every divisorial step. All connecting steps are therefore small, and their discrepancy comparisons preserve generalized terminality. They also preserve the exceptionality status of every place. Smallness downstairs keeps each marked place exceptional over X_{i+1}, whereas every other surviving prime corresponds to a prime on X_{i+1}. Thus the surviving marked primes are exactly the exceptional primes of h_{i+1}, and the end is the required junction Y_{i+1}. This proves the construction by induction. The relative program is allowed to have length zero.

There are finitely many marked primes and finitely many components of B_0. Smallness transports all of them through every connector. The downstairs coefficients are constant, and the marked coefficients are 1-a_i(E) at junctions and are changed only by (4.3). They are therefore nonincreasing and lie in [0,1). For the finitely many coefficient sequences that are eventually constant, discard a prefix after all have become constant. Every other sequence stays positive: a nonnegative nonincreasing sequence that reaches zero is thereafter zero. Only marked coefficients can be of this latter type. This proves the last assertion.□

Proof. We first prove the assertion about \mathcal A_{\mathrm{surf}}. If \mathcal I is empty, choose \mathcal A_{\mathrm{surf}}=\varnothing. For the remaining case, fix a junction \rho:Y\to X and a surface V\subset X which is the center of at least one marked prime. We work near general points of V.

A transverse surface and its curves. The exceptional image of a proper birational morphism to normal X has codimension at least two. We may avoid every other component of that image and every center properly contained in V. The full inverse image of V is a proper analytic subset of the irreducible fourfold Y, hence has dimension at most three. A component supplying a curve over a general point of V must therefore have dimension three and be a divisor dominating V. All such divisors are marked. Components of dimension at most two cannot supply a curve in a general fiber. These observations also show that the general fibers over V have dimension at most one.

The singular locus of terminal Y has dimension at most one, so it misses the fiber over a general point of V. The singular loci of the marked divisors, their pairwise intersections, and their intersections with strict transforms of components of B_0 have dimension at most two. Choose also a projective carrier \sigma:W\to Y for the fixed nef data, with W smooth. Its nonisomorphism image has dimension at most two. None of these subsets can contain an entire curve over a general point of V: a component that dominates the surface has zero-dimensional general fiber, and components with smaller image can be avoided.

Here is a precise choice of the slice used below. Near a general smooth point of V, lift two local coordinates on V to holomorphic functions on X. Take a general common level set of their pullbacks on Y. Finite analytic stratifications near the proper fiber can be chosen to respect the divisors and the subsets in the preceding paragraph. On every stratum dominating V, generic smoothness makes the two pulled-back functions submersive over general values; the images of all other strata can be avoided. It follows, after shrinking about the fiber, that the level set is a smooth surface R. The restriction of K_Y to R is K_R, because the two slice equations trivialize its normal bundle.

Let C_1,\ldots,C_q be the reduced irreducible curves of the fiber in R. Each is supplied by exactly one marked divisor, with multiplicity one in that divisor’s slice: at a general point of the curve the divisor is smooth, the slice is transverse on it, and none of the other listed subsets contains that curve. Write \lambda(j) for its label. The same label may supply several curves, but each marked divisor with center V supplies at least one. No C_j is contained in a strict downstairs boundary component or in the nonisomorphism image of \sigma.

The restricted map on the slice is proper near this fiber, contracts the C_j to a point, and is an isomorphism off that point locally. No surface component is mapped to the point, by the fiber dimension bound. Pass, if necessary, to the normal Stein factor of the local target, or equivalently to the normalization of the relevant local image. We then have a proper modification from the smooth surface whose reduced exceptional set is the union of these curves. Grauert’s negative-definiteness criterion [12, Section 4, no. 8(e), pp. 366–367] therefore applies: the matrix

is negative definite. Put u_j=-C_j^2\in\Z_{>0} and m_{jk}=C_j\cdot C_k\in\Z_{\geq0} for j\ne k.

Uniform bounds for the intersection equations. Let B_Y^{\mathrm{str}} denote the strict boundary part coming from B_i. It has nonnegative intersection with every C_j, since it is effective and contains no such curve. The same nonnegativity holds for M_Y. Indeed, the strict transform C_j' on W meets the isomorphism locus of \sigma. For the effective nef defect of Lemma 3.2,

the curve C_j' is not contained in \Supp J. The projection formula, nefness on the carrier, and effectivity then give

Intersect the crepant equation K_Y+\Theta+M_Y=\rho^*D_X with C_j. On R the marked part near the fiber is \sum_k b_{\lambda(k)}C_k. Adjunction for the reduced irreducible compact curve C_j gives K_R\cdot C_j=2p_a(C_j)-2+u_j. We obtain

Since p_a(C_j)\geq0 and 1-b_{\lambda(j)}\geq\varepsilon, it follows that

Negative definiteness now bounds each whole row of A, even when q is large. Fix j and test its quadratic form on the vector whose jth entry is 1 and whose kth entry is m_{jk}/u_k for k\ne j. Its value is

The last sum is nonnegative. Therefore

The second inequality uses u_k\leq U and the integrality and nonnegativity of every m_{jk}. The absolute sum in each row of A is thus at most C_{\mathrm{row}}:=U+U^2.

Grouping curves by global labels. Let \mathcal H_V be the marked labels whose center is V, and put t=|\mathcal H_V|\leq|\mathcal I|. Define the q-by-t zero-one matrix P by

Its columns have disjoint nonempty supports, so P has column rank t. Since A is invertible, AP has the same column rank. Each of its integer entries is a sum from one row of A, and hence has absolute value at most C_{\mathrm{row}}. The intersection equations take the form

No marked coefficient occurs in the right-hand side.

There is a fixed integer m_{\mathrm{data}}>0 such that every m_{\mathrm{data}}v_j is integral, for every junction and surface in this calculation. Choose it to clear the initial boundary coefficients of this strong generalized klt sequence and to make its chosen higher nef divisor integral Cartier. Canonical degrees on the smooth R are integral. Each strict boundary prime is Cartier in the smooth neighborhood of the fiber, so its degree has the chosen boundary denominator. Finally, the trace of the integral Cartier divisor m_{\mathrm{data}}\M on any model is an integral Weil divisor: compute it by pulling back to a common higher model and pushing forward. On the smooth neighborhood of the present fiber that trace is Cartier, so its degree is integral as well. No assertion about its Cartier index elsewhere is needed.

Choose t rows of AP giving a nonsingular square matrix. Its determinant is a nonzero integer with an absolute value bounded in terms of t and C_{\mathrm{row}}, for example by t^{t/2}C_{\mathrm{row}}^t by Hadamard’s inequality. Since 1\leq t\leq|\mathcal I|, choose one integer H_{\mathrm{det}} bounding all these absolute values. Cramer’s rule applied to (4.7) then gives

The denominator is independent of the junction, the surface, and the number of geometric curves in the slice. In particular the marked discrepancies above surfaces belong to the finite set

This finite set is used only to exclude recurring marked surface centers contained in the exceptional loci. The weight denominator in Section 6 will be chosen later from the actual fixed coefficients and the initial rational data; it need not clear every element of \mathcal A_{\mathrm{surf}}.

Removing marked surface centers from the exceptional loci. Fix E\in\mathcal I. Suppose its center is a surface in L_i at infinitely many indices i_1<i_2<\cdots. The strict comparison at i_k and monotonicity between indices give

Both a_{i_k}(E) and a_{i_{k+1}}(E) belong to \mathcal A_{\mathrm{surf}}, a contradiction. If instead its center is a surface in L_i^+ at infinitely many indices, the corresponding positive-side discrepancies satisfy

again contradicting membership of a_{i_k+1}(E) and a_{i_{k+1}+1}(E) in the same finite set. There are finitely many E\in\mathcal I, so one tail excludes all such occurrences on both sides. This proves assertion (1).

On that tail, suppose a marked prime has a surface center V at a junction over X_i. Assertion (1) says V is not contained in L_i. Its general point is therefore in the common isomorphism open of the next diagram. The next center is its strict transform, still a surface, and Lemma 3.4 gives equality of its two discrepancies. Repeating the argument at every subsequent index shows that its coefficient is constant from this index onward. A varying label cannot have such a surface center. Its center cannot be a divisor downstairs because it represents an exceptional place. This proves assertion (2).

Finally let V be an irreducible compact analytic surface in one of the exceptional loci of a remaining diagram, and use the junction on that side. If the junction were not an isomorphism at the general point of V, its general fiber there would be positive dimensional: a proper birational morphism to a normal space is an isomorphism wherever it is finite. A component supplying those curves over V would be a divisor with center V, by the dimension argument at the start of the proof. This is excluded by assertion (1). Thus the junction is an isomorphism near a general point of V. The source is ordinarily terminal and smooth in codimension two, so the downstairs model is smooth at a general point of this surface.

The coherent ideal of the global analytic surface V has a projective blowup. Over the dense open where the ambient model and V are smooth, this is the ordinary codimension-two blowup and has a unique prime divisor dominating V. Its closure on the normalized blowup defines a global place with center V, hence an exceptional place on the downstairs model. If its discrepancy were at most 1, it would belong to the stabilized set \mathcal I, contradicting assertion (1). This proves assertion (3).□

Proof. For a reduced pure projective threefold R, write \nu_R\colon N_R\to R for normalization. Choose a closed reduced subset A_R\subset R of dimension at most two containing the locus where \nu_R is not an isomorphism, and put F_R=(\nu_R^{-1}A_R)_{\mathrm{red}}. The square

is a proper excision square. Indeed \nu_R is finite and proper and identifies the open complements. Collapsing the two closed subsets gives a continuous bijection N_R/F_R\to R/A_R of compact Hausdorff quotients, hence a homeomorphism. Comparing the two relative homology sequences gives the exact sequence

All its maps are compatible with mixed Hodge structures. This follows from algebraic functoriality and the exact sequence of relative cohomology as a sequence of mixed Hodge structures [7, Propositions 8.2.2 and 8.3.9], followed by duality on these compact spaces.

Take this square first for P_1. For Q, take A_Q=Q\cap A_{P_1}. The normalization N_Q=\coprod_jQ_j^\nu is a union of entire, disjoint normalization components of N_{P_1}. Thus these choices give a morphism of the two squares, with F_Q equal to the part of F_{P_1} in those selected components.

For a surface component V of A_Q, let F_{V,1},\ldots,F_{V,r_V(Q)} be the surface components of F_Q mapping onto it, and let d_{V,\alpha}>0 be their finite degrees over V. On top homology the corresponding part of the map to A_Q is

Its kernel has dimension r_V(Q)-1. A surface not contained in A_Q has exactly one normalization divisor over it, since the normalization is an isomorphism at its general point. Conversely, finiteness and surjectivity of normalization give at least one divisor over every surface in Q. It follows that

This also proves finiteness of the sum: the only possible nonzero terms come from the finitely many surface components of A_Q.

The image of B_Q in H_4(N_Q) is spanned by classes of compact surfaces in its normalization components. Its rank is therefore at most \sum_jc_2(Q_j^\nu). Hence

By (5.6), U_Q is the image of \partial_Q.

The map H_4(F_Q)\to H_4(F_{P_1}) is injective. Both groups have surface components as their bases, and the surfaces in F_Q remain distinct surfaces on the selected normalization components of P_1. Naturality of the boundary maps therefore sends U_Q injectively into the image of \partial_{P_1} on the image of H_5(Q)\to H_5(P_1). The groups H_4(F_Q) and H_4(F_{P_1}) are pure of weight -4. Strictness for the weight filtration [6, Theorem 2.3.5(iii)–(iv)] says that \partial_R maps \Gr^W_{-4}H_5(R) onto its image in the pure group H_4(F_R). Applying this to the naturality square, we obtain

Finally, Borel–Moore localization contains

The first group vanishes because P\setminus P_1 has complex dimension at most two. The second arrow is therefore injective, and remains injective on weight-graded pieces by strictness. Duality between rational homology and cohomology reverses the sign of the weight and turns the last two inequalities into (5.5).□

Proof. Write B for the target of f and K_f=Rf_*\Q_{\widetilde Y}[4]. Fix an f-relatively ample line bundle and let \ell be its rational first Chern class. Cup product defines a global morphism \ell\colon K_f\to K_f[2].

We first obtain relative hard Lefschetz locally on B. Around a point of B, choose a small chart V embedded as a closed analytic subspace of a complex manifold. After making the chart smaller, relative ampleness embeds f^{-1}V in a projective space over V; composing with the closed embedding of V gives a projective map from the smooth manifold f^{-1}V to the ambient manifold. On each connected component of the smooth source, the constant shifted sheaf underlies the constant polarizable Hodge module with that strict support. Saito’s projective direct-image theorem [23, Theorem 5.3.1] applies to precisely this situation: its morphisms are projective morphisms of smooth complex analytic manifolds, its Lefschetz class is the Chern class of a relatively ample line bundle, and its conclusion includes relative hard Lefschetz and polarizable perverse direct images. Passing through the closed embedding, which is fully faithful and exact for perverse sheaves, gives on V

After forgetting Hodge structures, the Tate twist has the same underlying rational perverse sheaf. The operator is the restriction of the single global \ell, and a morphism of perverse sheaves is an isomorphism if it is so on an open cover. Thus these are global relative hard Lefschetz isomorphisms on B.

The formal Lefschetz splitting criterion [5, Theorem 1.5] now gives

The criterion uses truncation triangles and the induced Lefschetz isomorphisms; the same argument applies to the bounded perverse t-structure. This explains why the local analytic theorem gives a global derived splitting here.

We also need to separate the supports of the perverse terms in (5.8). Locally, Saito’s pure direct images decompose by strict support [23, Section 5.1 and Theorem 5.3.1]; their underlying perverse terms are middle extensions from smooth opens of those supports. The support decomposition glues without requiring semisimplicity of local systems on arbitrary analytic opens. Indeed, a perverse morphism between middle extensions with different irreducible supports must be zero: its image would be a subobject or a quotient with smaller support in at least one of them. Restriction to an open preserves the middle-extension property; every resulting strict support is a local branch of the original support. The local system may decompose further without changing that support. For each perverse term and each dimension d, the sum of the local projectors onto all d-dimensional strict-support summands is intrinsic. Restriction preserves dimensions of local branches, so these sums agree on overlaps and glue as perverse morphisms. The support of each glued dimension summand is closed analytic in every chart, hence globally. For each irreducible component Z of this support, sum all local projectors whose strict supports are local branches of Z. These sums agree by the same zero-morphism argument and glue to a global projector whose image has strict support Z; the middle-extension property is local. Local finiteness and compactness make the number of these summands finite.

Finally f is an isomorphism over a dense open of its target. On that open K_f is \Q[4] in perverse degree zero. The only full-support term is consequently the middle extension of this constant local system, namely \IC_B, in degree zero. This proves the lemma.□

Proof. Apply Lemma 5.3 to f=pg. Besides \IC_X, write a strict-support summand as \mathcal K_{Z,i}[-i], where \mathcal K_{Z,i} is perverse with proper irreducible support Z\subset X of dimension z. Over a general point x of Z, the fiber of f has dimension at most 3-z: f^{-1}(Z) is a proper analytic subset of the irreducible fourfold \widetilde Y and therefore has dimension at most three. Proper base change and the topological dimension of a compact analytic fiber give

The generic nonzero stalk of \mathcal K_{Z,i}[-i] is in degree i-z, because a perverse middle extension on Z has its generic local system in degree -z. It is a direct summand of this stalk, so

We now restrict the decomposition to C. Suppose first that Z\not\subset C, and stratify Z\cap C compatibly with \mathcal K_{Z,i}. A stratum S has dimension s\leq z-1 and s\leq1. Its stalk degrees for \mathcal K_{Z,i} are at most -s-1 by (5.7). This remains valid under the refinement by C: a stratum cut out of an old boundary stratum inherits the stronger bound from that stratum, while a stratum cut out of the local-system open has only degree -z\leq-s-1. Compactly supported cohomology on the smooth complex s-fold S vanishes above degree 2s. Thus its contribution to the hypercohomology of \mathcal K_{Z,i}[-i]|_C vanishes in degrees greater than

Filtering C by its finitely many strata proves that this summand has no degree-one hypercohomology on C. The full-support summand \IC_X has the same property: for each stratum of C, the upper bound is -s-1+2s=s-1\leq0; on the smooth open its only stalk degree is -4, which is smaller still.

Only summands with Z\subset C can therefore contribute in degree one. Such a complex is supported on the closed subset C, so its restriction induces an isomorphism on hypercohomology from X to C. The chosen global splitting and proper base change now show that

is surjective.

It remains to justify what this surjection says about weights. Compact Kähler Hodge theory gives H^5(\widetilde Y,\Q) its pure Hodge structure of weight 5. On the projective \widetilde P, Deligne’s mixed Hodge structure has weights at most 5 [7, Theorem 8.2.4(iii)]. We verify directly that the restriction map preserves its Hodge filtration, because the ambient \widetilde Y may be nonprojective.

Choose a proper smooth projective hyperresolution \epsilon_\bullet\colon Z_\bullet\to\widetilde P. Deligne’s construction computes the Hodge filtration on H^5(\widetilde P) from the total simplicial de Rham complex of the Z_n, filtered by holomorphic form degree [7, Section 8.1, (8.2.1), and Proposition 8.2.2]. The terms can be chosen projective because \widetilde P is projective and the construction can use projective resolutions and modifications. We may use smooth forms in this complex, with the same filtration: the Dolbeault resolutions give filtered quasi-isomorphisms from the holomorphic de Rham complexes on the smooth Z_n. A class in F^aH^5(\widetilde Y,\C) has a closed smooth representative \eta whose components have holomorphic degree at least a, by the Kähler Hodge decomposition. Pullback along the holomorphic map Z_0\to\widetilde P\hookrightarrow\widetilde Y preserves these bidegrees. Put that pullback in simplicial degree zero of the total complex. Its de Rham differential vanishes, and its simplicial differential is zero because the two face pullbacks to Z_1 have the same composite with the augmentation. It is therefore a cocycle in the filtered total de Rham complex. It represents the ordinary topological restriction by de Rham naturality and cohomological descent. Thus restriction preserves F^a for every a.

The restriction is rational on underlying cohomology. It also preserves W: the source is pure of weight 5, and W_5 is all of the target. It is consequently a morphism of mixed Hodge structures. Strictness for W [6, Theorem 2.3.5(iii)], applied to the surjection just proved, yields

Together with the upper weight bound, this proves purity of weight 5. Strictness here concerns mixed Hodge structures themselves and does not require a rational polarization on the compact Kähler source.□

Proof. Use the full-support projection in the chosen decomposition for g to form the morphism

The first arrow is the natural pullback map. Both arrows restrict to the identity on Y_{\mathrm{reg}}. Let \mathcal L be the cone of \alpha, so that

It is supported on \Sing Y. Enlarge the finite puncture sets, if necessary, so that the cohomology sheaves of \mathcal L are local systems on each T^\circ. Analytic constructibility permits this enlargement, and the preceding coinvariant description shows that it does not change b_T.

By proper base change, the map induced by \alpha on P factors as

The first arrow comes from a holomorphic morphism between the projective spaces \widetilde P and P, and is algebraic by GAGA. It therefore preserves weights. Lemma 5.4 shows that it kills W_4H^5(P). Hence the displayed composite kills that subspace as well; we need no weight statement about the chosen perverse projection. The long exact sequence of (5.12) on P gives

Restricting the triangle from P to Q shows that the image of this subspace in H^5(Q) is contained in the image of \mathbb H^0(Q,\mathcal L|_Q). Passing to a weight-graded quotient can only lower rank, and therefore

We estimate the last group by its strata. On a stratum of \Sing Y of dimension s\in\{0,1\}, the bound (5.7) and the triangle give

The constant complex in the triangle has its only stalk cohomology in degree -4; its shifted contribution to the cone is in degree -5, so it cannot alter this upper bound. On a curve T^\circ, the same triangle identifies \mathcal H^{-2}(\mathcal L) with \mathcal H^{-2}(\IC_Y). The curve stratum has complex codimension three in Y. In the middle-extension construction, attaching a codimension-c stratum retains the unshifted cohomology of Rj_*\Q through degree c-1. For c=3 this gives

This is the attaching truncation of [11, Section 3.1 and Section 3.3, Axiom [AX1](d)], with the shift by four from our perverse normalization.

The restriction \mathcal L|_Q is supported on the closed set \Sing Y\cap Q, so its hypercohomology may be computed on that set. The complement there of the disjoint opens T^\circ for the curves contained in Q is finite. A point stratum contributes nothing to degree-zero hypercohomology by (5.14). The open–closed cohomology sequence therefore gives a surjection

On a curve, compactly supported cohomology of a local system vanishes above degree two. In the spectral sequence

the stalk bound gives b\leq-2. The only term in total degree zero is consequently a=2, b=-2, whose dimension is b_T by (5.15). The dimension of each degree-zero abutment is at most b_T. This bound and (5.13) prove (5.11).□

Proof. The isomorphism of g over the regular locus and adjunction give a natural morphism

By the definition of a higher direct-image stalk and proper base change for g, its degree--2 map at t is precisely (5.18). These are the usual sheaf cohomology groups of constant sheaves, equal to singular cohomology on the locally contractible analytic spaces in question.

Let \kappa\colon\IC_Y\to Rg_*\Q_{\widetilde Y}[4] be the full-support inclusion in the chosen decomposition. Its restriction to Y_{\mathrm{reg}} is the identity. The composite \rho\kappa\colon\IC_Y\to Rj_*\Q[4] is therefore the canonical comparison morphism: by adjunction a morphism to Rj_*\Q[4] is determined by its restriction to Y_{\mathrm{reg}}. The codimension-three attaching construction used in (5.15) says that this comparison is an isomorphism on ordinary stalk cohomology in degree -2 along T^\circ. Its degree--2 map factors through

Thus the degree--2 map induced by \rho, namely (5.17), is surjective. It is a morphism of the local systems in (5.10), which gives the asserted monodromy equivariance.□

Proof. Fix T, and write

for the natural quotient of Lemma 5.6. If b_T=0 there is nothing to prove, so assume b_T>0. We may remove finitely many further points of T^\circ whenever needed. The local systems already defined restrict to the smaller curve, and the induced map on fundamental groups is surjective. Thus their monodromy images and the coinvariant dimension b_T do not change.

We first construct sections

whose images under q_T are linearly independent in every fiber of \mathcal V_T, after the permitted further finite puncturing.

Step 1. Invariant lifts from finite monodromy. First we show that \mathcal W_T, and therefore its quotient \mathcal V_T, has finite monodromy. Let \mathcal F=\widetilde Y\times_Y T, with its full complex-space structure. It is projective over the compact projective curve T. A relative embedding and GAGA identify this morphism with an algebraic projective family, including its possibly nonreduced scheme fibers. After deleting finitely many points we may assume that \mathcal F\to T^\circ is flat and that all (R^kg_*\Q)|_{T^\circ} are local systems.

For t\in T^\circ, put F_t=\mathcal F_t. Every integral class in H^2(F_t,\Z) lifts, by proper base change, to H^2(g^{-1}U_t,\Z) for a sufficiently small Stein neighborhood U_t of t in Y. Terminal singularities are rational [10, Remark 3.8 and Theorem 3.12]. Hence g_*\mathcal O_{\widetilde Y}=\mathcal O_Y and R^kg_*\mathcal O_{\widetilde Y}=0 for k>0; Leray and Cartan’s Theorem B give

The exponential sequence therefore realizes the lifted integral class as the first Chern class of a holomorphic line bundle on g^{-1}U_t. Its restriction to the projective scheme F_t is algebraic by the scheme form of GAGA [22, Corollary 4.3 and Theorem 4.4]. In particular,

Here and below the cohomology of a scheme fiber means that of its analytic topological space; nilpotents do not change it.

We use the parameter argument of [20, Section 6.1, proof of Lemma 6.4] to show that a spanning set of fiber classes has finite monodromy orbits. Fix a relatively very ample \mathcal O_{\mathcal F}(1). Every line bundle on a fiber is a quotient of some \mathcal O_{F_t}(-n)^{\oplus r}. Letting n,r, and the Hilbert polynomial vary gives countably many relative Quot schemes of finite type over T^\circ [13, Theorems 3.1–3.2]. In each take the open set on which the universal quotient is invertible on the whole pulled-back family. These open sets still contain a point representing every fiber line bundle. Indeed the family and universal quotient are flat over the Quot scheme, so the local criterion for flatness makes a fiberwise locally free quotient of rank one locally free near that fiber; properness removes the image of the complementary closed locus.

There are countably many irreducible components of these parameter spaces. Choose t outside the closures of the images of all components which do not dominate T^\circ. Such a point exists, since each of those closures is a finite set and a complex curve is uncountable. Put \Gamma=\pi_1(T^\circ,t). By (5.19), choose finitely many line bundles on F_t whose Chern classes span (\mathcal W_T)_t. Each is represented on a dominating reduced irreducible parameter component H. The Chern class of the universal bundle gives a section of the pullback of \mathcal W_T to H^{\mathrm{an}}: restrict its class on the total family to fibers and use proper base change.

Choose a closed point of the generic fiber of H\to T^\circ. Its residue field is a finite extension of the function field of T. Spreading this point, normalizing its closure, and deleting finitely many base points gives a connected finite étale cover of a dense open of T^\circ, together with a morphism from that cover to H. The analytification of an irreducible complex algebraic variety is path connected. A path in H^{\mathrm{an}} from the original parameter point to a point of this cover identifies the transported Chern class with the class there. The subgroup fixing that sheet has finite index and fixes this class. The inclusion of the further punctured curve, with the projected path for change of basepoint, still surjects on \Gamma. The image of the finite-index sheet stabilizer is therefore finite index in \Gamma and fixes the original class. Thus each of our spanning classes has a finite orbit under \Gamma. The intersection of their stabilizers has finite index and acts trivially on (\mathcal W_T)_t. The monodromy group G of \mathcal W_T is therefore finite, as is that of \mathcal V_T.

Poincaré duality with local coefficients on the oriented curve gives

For a representation of the finite group G over \Q, averaging identifies coinvariants with invariants and makes invariants an exact functor. The quotient q_T therefore supplies the asserted sections \sigma_i with independent images.

Step 2. Line bundles near a compact curve core. Take T^\circ noncompact, removing one more point if needed. Recall the fixed choice made after (5.10): for every prime component E of \Exc(g) dominating T, its Stein factor restricts to

where the first map is a smooth proper family with connected fibers and the second is a connected finite étale cover. Further finite puncturing preserves these properties. Choose an ambient open Y^\circ\subset Y in which T^\circ is closed: remove the discarded points of T, the other singular components, and their intersection points. These choices will be used when we identify the local divisors globally.

Let h:g^{-1}T^\circ\to T^\circ be the restricted resolution. The curve T^\circ has the homotopy type of a finite graph, so cohomology of its local systems vanishes in degrees at least two. The Leray spectral sequence for h consequently makes the edge map

surjective. Choose lifts of the \sigma_i. The algebraic space g^{-1}T^\circ has finite triangulation type; after multiplying each lift by a positive integer, it comes from a class \alpha_i\in H^2(g^{-1}T^\circ,\Z). Replace \sigma_i by the same multiple. Their images are still independent.

To extend the \alpha_i simultaneously while retaining every loop of T^\circ, choose a compact curve core as follows. In the smooth compactification of T^\circ, remove small open coordinate discs about its punctures, with disjoint closures, and denote the complement by K_*. This is a connected compact semianalytic subset of T^\circ. The inclusion of its interior into T^\circ induces a surjection on fundamental groups. The core K_* is also Runge: every component of T^\circ\setminus K_* runs to a missing puncture and is therefore not relatively compact. The compact set g^{-1}K_* is semianalytic. Continuity of cohomology on neighborhoods of this compact set extends the restrictions of the finitely many \alpha_i to integral classes on one open neighborhood \widetilde U_0 of g^{-1}K_* in \widetilde Y. One may use Čech cohomology for this continuity statement; semianalytic triangulation and local contractibility identify it with singular cohomology here. Properness of g makes

an open neighborhood of K_* with g^{-1}U_0\subset\widetilde U_0.

The closed curve T^\circ\subset Y^\circ is Stein. Siu’s Stein-neighborhood theorem, in the form [8, Theorem 2.1] with empty compact set, gives a Stein neighborhood \Omega of this curve in Y^\circ. The Runge property makes K_* holomorphically convex in T^\circ, hence also in \Omega by the paragraph preceding that theorem. The usual holomorphic-polyhedron construction for a convex compact in a Stein space gives a Stein neighborhood

Take its connected component containing K_*. It is normal, being open in Y, and therefore irreducible. For the local \Q-factorialization, Fujino’s compact construction [10, Section 1.11] supplies an ambient semianalytic Stein compact K'\Subset U, with K_*\subset\operatorname{int}_U(K'), such that

for every analytic subset Z defined on a neighborhood of K'. This is the compact condition (P4) in Fujino’s condition (P); normality of the source and Steinness of the base supply its other ambient conditions.

The classes just extended to g^{-1}U are Chern classes of line bundles L_1,\ldots,L_{b_T} there, by the same rational-singularity, Cartan, and exponential-sequence argument used above. For every t in the interior of K_*, their fiber classes are the \sigma_i(t); hence their images under q_T remain independent. We also need divisor representatives before making any further modification. The sheaf g_*L_i is coherent and has rank one on the regular open of U. Cartan’s Theorem A supplies a global section nonzero at a point of that open. Its pullback is a nonzero section of L_i, whose zero divisor is a Cartier divisor A_i on g^{-1}U representing L_i.

Step 3. Fiber degrees and surface families. We will obtain at least b_T surface families by proving that the b_T line bundles constructed below have linearly independent vectors of fiber degrees. Apply Fujino’s small \Q-factorialization theorem [10, Theorem 1.24] to the ordinary pair (U,0), the identity U\to U, and the full compact K'. The identity is projective, U is terminal and therefore klt, and K_U is \Q-Cartier. The preceding construction supplies condition (P). After restricting the base to a neighborhood of K', the theorem gives a projective small bimeromorphic morphism

with Y' normal and \Q-factorial over K'. We retain the notation U after this restriction and restrict g,L_i,A_i accordingly. Compactness and local terminality of Y give a common m_{\mathrm{can}}>0 for which \omega_Y^{[m_{\mathrm{can}}]} is invertible; its restriction to U is \omega_U^{[m_{\mathrm{can}}]}. Since \ell is small, the natural identification on the common codimension-one open extends by reflexivity to

In particular, the left side is invertible and the natural relative canonical divisor K_{Y'/U} is zero. This is canonical crepancy, expressed using compatible local canonical generators as K_{Y'}=\ell^*K_U. A place exceptional over Y' is exceptional over U, because \ell is small. Crepancy therefore preserves the terminal discrepancy inequalities, so Y' is terminal. Moreover \ell is an isomorphism over U_{\mathrm{reg}}. Indeed, for a relatively ample line bundle \mathcal A on Y', the rank-one reflexive sheaf (\ell_*\mathcal A)^{**} is locally invertible on the smooth base. Its pullback agrees with \mathcal A off codimension two and hence everywhere there, by reflexivity on the normal source. A positive-dimensional fiber over a smooth point would then have degree zero against \mathcal A, contradicting relative ampleness.

Push A_i forward to a Weil divisor on U, and let D_i' be its strict transform on Y'. These are actual locally finite analytic divisors defined on all of the current Y'. We apply \Q-factoriality precisely in the sense of [10, Definition 2.38]: every prime divisor defined on a neighborhood of the whole set \ell^{-1}(K') is \Q-Cartier at each point of that set. Properness makes \ell^{-1}(K') compact, and local finiteness leaves only finitely many prime components of the D_i' near it. A finite cover of this full compact and a common multiple therefore give an integer m_{\mathrm{div}}>0 and an open neighborhood of \ell^{-1}(K') on which every m_{\mathrm{div}}D_i' is Cartier. By properness, choose an open set U_1 with K'\subset U_1\subset U whose full inverse image is contained in that neighborhood, and put

This completes the use of \Q-factoriality on the specified full compact; all later restrictions use these fixed line bundles. In particular, the change-of-compact issue in [10, Remark 2.39] does not arise. On the regular open, where both g and \ell are isomorphisms,

Choose connected curve domains

such that \pi_1(B_0)\to\pi_1(T^\circ) is surjective; nested smaller cores obtained by enlarging the omitted discs have this property. The larger domain lets us control finitely many components near \overline B_0 and make their discrete exceptional sets finite on that compact. Work in the ambient open of U_1 obtained by removing T^\circ\setminus B_1, so that B_1 is closed there. The inverse image of B_1 is proper over B_1. Local finiteness of its analytic components, compactness of \ell^{-1}(\overline B_0), and properness allow us first to replace B_1 by a neighborhood of \overline B_0 on whose inverse image only finitely many components occur.

We claim that deleting finitely many points from B_0 gives a connected curve domain B with \pi_1(B)\to\pi_1(T^\circ) surjective and the following fiber description. All fibers of \ell over B have dimension at most one. The finitely many irreducible surface components S_1,\ldots,S_s of the reduced inverse image of B, counted after this restriction, all dominate B. Their normalizations factor as

where C_\alpha\to B is a connected finite étale cover and h_\alpha is a smooth proper family of connected curves. A cover that splits on restriction contributes separate components to this list. For each fixed t\in B, the pairs (\alpha,c) with c\in C_\alpha over t supply distinct irreducible curves and exhaust the irreducible curves of \ell^{-1}(t). Each smooth connected normalization fiber maps birationally to the curve it supplies.

We verify this description by spelling out the finite exclusions. Smallness gives \dim\Exc(\ell)\le2, so a general fiber above B_1 has dimension at most one. The locus of larger fibers is a proper analytic subset of B_1, by the fiber-dimension theorem for proper maps. For each surface component of the reduced inverse image that dominates B_1, take its normalization and then its Stein factor over the curve. The normalization is proper and is a normal surface. Its singularities are isolated. Away from the branch values of the finite Stein factor and the critical values of the map from the normalization, it is a smooth proper family of connected curves over a finite unramified cover. The values at which an intersection of two components, or the nonisomorphism locus of a normalization, contains a whole fiber curve also form a discrete analytic subset: these loci have dimension at most one, and their general fibers over the base are zero-dimensional. Finally remove the images of components lying over a point. Each of these excluded sets is discrete on the larger curve domain B_1; there are only finitely many components, so their union meets \overline B_0 in a finite set.

Delete those finitely many points from B_0 and call the result B. It is connected, and \pi_1(B)\to\pi_1(T^\circ) is still surjective: loops in B_0 may be perturbed away from a finite set. The exclusions give the asserted fiber description and families.

For each \alpha,i, let

where pullback to \widehat S_\alpha is understood. This integer is independent of c, by constancy of degree in a smooth proper family over the connected curve C_\alpha. Because its fiber maps birationally to the curve it supplies, this is also the degree of \mathcal N_i on that fiber curve of \ell. We claim that the map

is injective.

Suppose its value is zero, and fix t\in B. Every irreducible curve in F'_t=\ell^{-1}(t) then has degree zero against \sum_i c_i c_1(\mathcal N_i). The fiber has dimension at most one. Its rational H_2 is spanned by the fundamental classes of its irreducible curves, also when the fiber is nonreduced, and H^2(F'_t,\Q) is the rational dual of H_2(F'_t,\Q). The restricted Chern class is therefore zero. Proper base change identifies it with the germ in (R^2\ell_*\Q)_t. Restriction through the isomorphism over the regular open gives the natural map

so the corresponding punctured class is zero as well. By (5.21), the restrictions of \sum_i c_i c_1(\mathcal N_i) and m_{\mathrm{div}}\sum_i c_i c_1(L_i) agree on their common regular open. Taking their germ in (R^2j_*\Q)_t and applying (5.18) to the latter identifies that zero class with

These vectors are independent; hence every c_i=0. This proves (5.22), and in particular

Step 4. Distinct global discrepancy-two divisors. Take an ambient open U_B\subset U_1 whose intersection with T^\circ is the closed subspace B, obtained as above by removing T^\circ\setminus B. Each S_\alpha is a closed analytic surface in \ell^{-1}U_B. Since Y' is terminal, it is smooth in codimension two by [21, Lemma 2.3]. Thus at a general point of S_\alpha both the ambient space and the surface are smooth. Blow up its coherent ideal and normalize. The unique divisor dominating that dense smooth part defines a local place v_\alpha of ordinary log discrepancy 2 over Y', the value for a smooth codimension-two blowup. Canonical crepancy of \ell gives

The v_\alpha are distinct because their centers on Y' are the distinct S_\alpha.

It remains to represent each v_\alpha by a prime divisor on g^{-1}U_B and show that these local divisors arise from distinct global components of \Exc(g).

On the fixed global resolution, the relative canonical divisor is intrinsic even if the canonical bundle has no global divisor representative. Using the fixed m_{\mathrm{can}}, divide by m_{\mathrm{can}} the divisor of the natural meromorphic comparison g^*\omega_Y^{[m_{\mathrm{can}}]}\dashrightarrow \omega_{\widetilde Y}^{\otimes m_{\mathrm{can}}}, and denote the result by K_{\widetilde Y/Y}. It is exceptional; write

The natural canonical comparisons commute with restriction and composition, so this divisor supplies the relative discrepancy term on U_B as well. Every \kappa_E is strictly positive by terminality. The principalization of the ideal of \Sing Y says that g^{-1}(\Sing Y) is supported on these exceptional divisors. Put \widetilde U_B=g^{-1}U_B, and consider v_\alpha on a common proper model with \widetilde U_B. If it were exceptional over this smooth resolution, then a(v_\alpha;\widetilde U_B,0)\ge2. Its center on the resolution maps into B\subset\Sing Y, so it is contained in at least one of the E, with \ord_{v_\alpha}(E)>0. The discrepancy formula would give

a contradiction. Thus v_\alpha is represented by a prime divisor on g^{-1}U_B. Since U_B\cap T=B, this local divisor is an irreducible component of E|_B for a global exceptional divisor E with g(E)=T.

Such a global divisor cannot split into two components over B. Indeed the connected finite cover \widehat T_E^\circ\to T^\circ in (5.20) remains connected over B: the action of \pi_1(T^\circ) on a fiber is transitive, and \pi_1(B) has the same image. Over this connected cover the restriction of E is a smooth proper family with connected fibers, so its total space is connected and smooth, hence irreducible. Therefore the distinct local divisors representing the v_\alpha come from distinct global components E of \Exc(g). Their global centers are T, and their ordinary log discrepancies are 2, since this equality holds on the open over B. Together with (5.23), this proves the proposition.□

Proof of Theorem 5.1. Every global place of ordinary log discrepancy 2 whose center is a singular curve is represented by a component of \Exc(g). Indeed the same discrepancy computation in the last part of Proposition 5.7 would otherwise make its discrepancy strictly greater than 2. There are only finitely many such components on the compact resolution, so \tau_Y(Q) is finite. Proposition 5.7, summed over the singular curves contained in Q, gives

Combining this with (5.16) yields

as required.□

Proof. Take a projective resolution \pi:W\to Y with W smooth and carrying \M, with divisorial exceptional locus, and resolving all labels. Further blowups of intersections in the simple normal crossing divisor make the strict transforms of the labels smooth and pairwise disjoint, while keeping the full divisor simple normal crossing. Write

Every \pi-exceptional component of \Theta_W has coefficient 1-a(E;Y,\Theta+\M)<0, by generalized terminality. The only possible positive components of \Theta_W are therefore the strict transforms of the positive labels. On higher models the nef datum pulls back from W, so the discrepancy calculation is the ordinary one for the signed boundary \Theta_W.

Consider a place E that is also exceptional over W, and let its center on W have codimension c\geq2. At a general smooth point of this center choose normal parameters x_1,\ldots,x_c. The Jacobian formula on a smooth model carrying E gives

Indeed, in the pullback of a local volume form at most one differential factor can lose one order along E; adding one to the Jacobian order gives the displayed bound. If the center lies in no positive component of \Theta_W, the signed boundary can only raise this discrepancy, which is at least two. Otherwise it lies in a unique positive smooth label of coefficient b<1. Taking x_1 as its local equation and discarding the negative boundary terms gives

A discrepancy less than two thus forces c=2 and containment in a positive label. A label of coefficient zero cannot give such a place.

If this codimension-two center is contained in \Exc(\pi), it is a component of the intersection of its label with one of the finitely many exceptional divisors. There are only finitely many such centers. If it meets the isomorphism open, its general point maps isomorphically to a surface downstairs, and the center is the strict transform of that surface. Thus a center mapping to codimension at least three belongs to the first finite list, and for any fixed surface downstairs there are only finitely many possible centers on W.

There are only finitely many places of discrepancy less than two whose center on W equals a fixed one of these codimension-two centers. To see this, discard the negative components of \Theta_W; this lowers discrepancies, so it suffices to work with the single smooth label of coefficient b. Blow up the center generically. The new exceptional coefficient is -(1-b). A further place with discrepancy less than two has its center in that exceptional divisor; the estimate (6.4) also forces a codimension-two center in the strict label. It must therefore follow their intersection above the general point of the original center. At stage j the most recent exceptional coefficient is -j(1-b), and the older exceptional divisors do not meet this generic intersection. A place above it that is still exceptional has discrepancy at least

This reaches two after finitely many stages. The same argument shows that, where no other boundary or nef data enter, the places below two are exactly the echoes in (6.2).

This generic computation identifies global places. At each global blowup there is a unique irreducible component of its exceptional divisor that dominates the dense smooth part of the center. The global analytic intersection of that prime with the strict label then has a unique irreducible component dominating the same part. These global components have coherent ideals, which define the next global blowups. Conversely, one can follow any given global place on common higher models through these blowups; when its center becomes a divisor on a normal model, that divisor represents the place. Thus the computation does not count different local pieces of one place separately.

Finally include the finitely many divisors on W that are exceptional over Y. The complement of the isomorphism open, and the loci where a label fails to be smooth and alone or the nef datum fails to descend, are contained in a proper analytic subset of Y of codimension at least two. Such a subset has only finitely many surface components on the compact fourfold. Take these components for \mathcal V_Y, enlarging the set by the finitely many surface images already encountered if necessary. Outside them the preceding one-label calculation applies, or there is no positive label and no place of positive weight. This proves all assertions.□

Proof. Write Y=Y_i and let Q=\bigcup_{h\text{ varying}}S_h with its reduced structure. If Q is empty, (6.8) makes every surface bracket nonnegative, and the other terms of (6.7) are nonnegative. Suppose Q\ne\varnothing. At this junction the map h_i is projective bimeromorphic from an ordinarily terminal compact Kähler fourfold, and \dim h_i(Q)\leq1 by Proposition 4.2. Thus Theorem 5.1 applies. For a surface V\subset Q, its branch count in that theorem is r_V(Q)=r_V^{\mathrm{var}}\geq1; the last inequality follows from the finite surjective normalizations. Surfaces outside Q have r_V^{\mathrm{var}}=0. The total possible deficit in (6.8) is consequently \beta(Q).

Every varying label has d_h\geq1. The places counted by \tau_Y(Q) in Theorem 5.1 also each supply a weight of at least one in the first sum of (6.7). In fact such a place E has ordinary log discrepancy two and center a singular curve in Q. It is exceptional over Y. Its center is contained in a varying label of positive coefficient, and that effective label is \Q-Cartier on the strong model. Its order at E is therefore strictly positive. On a common carrier the discrepancy formula reads

The count \tau_Y(Q) uses distinct global places. They occur as distinct terms of the codimension-at-least-three sum, and none is a place in a surface bracket. Theorem 5.1 now gives

Summing (6.8) and substituting this inequality in (6.7) proves the assertion. The branch inequality and this lower bound have been used only at the junctions.□

Proof. Let T_h be the normalization of the common image of the two labels in Z. Restricting the projective maps, composing with finite normalization, and factoring through T_h gives projective bimeromorphic maps of normal compact threefolds

They identify over the common isomorphism open. By Lemma 3.4, the common exceptional image A=f(L)=f^+(L^+) has dimension at most one. A normalization divisor lying over L has a surface as its image in S_h, because normalization is finite, and its image in T_h lies over A. It is therefore contracted by the displayed map. Conversely, the map is an isomorphism off the preimage of A, so every contracted prime divisor lies over L. These are exactly the primes counted in the first sum in (6.11). The same reasoning holds on the positive side. The exceptional loci have finitely many surface components, which also proves finiteness.

We use the following independence fact. For a projective bimeromorphic map g:S\to T of normal compact threefolds, the classes in H_4(S,\Q) of the prime divisors contracted by g are linearly independent. Take a resolution r:\widetilde S\to S that is projective, has smooth source, and is an isomorphism over U=S_{\mathrm{reg}}. Put R=r^{-1}(S\setminus U). Normality gives \dim(S\setminus U)\leq1, and \dim R\leq2. If a rational combination of the contracted divisor classes vanished on S, the corresponding combination of strict transforms would restrict to zero in H_4^{\BM}(U,\Q). The localization sequence

then shows that its class is a rational combination of the classes of the surface components of R; those classes span H_4(R,\Q). Subtract that combination. We obtain a homologically trivial rational Cartier divisor on \widetilde S, all of whose components are exceptional over T. For the components of R, this follows since their images in S, hence in T, have dimension at most one. On the smooth compact resolution, homological triviality makes this divisor numerically trivial on every curve by the ordinary class pairing. Analytic negativity in both signs, in the form of [21, Lemma 2.2], makes the divisor zero. The strict transforms of the original divisors are distinct from the resolution-exceptional ones, so all original coefficients vanish. This proves independence without any factoriality assumption on S.

For either label normalization, restrict its span of compact analytic surface classes to the common open over T_h\setminus\nu^{-1}(A), where \nu:T_h\to f(S_h) is normalization. The complement of this open in either normalization has dimension at most two. Localization shows that the kernel of this restriction is spanned by its surface components, precisely the contracted prime divisors just counted. Their independence makes the kernel dimension e_h on one side and e_h^+ on the other. The two images agree: any compact analytic prime surface not removed has a strict transform on the other normalization through their proper graph over T_h, and the two classes restrict to the same class on the common open. The same holds in the reverse direction. Subtracting the dimensions of these two cycle spans proves the last equality in (6.11).□

Proof. For a coefficient change, the removed boundary \Theta-\Theta' is effective and \Q-Cartier on the strong model. For every place,

Thus w(E)\geq w'(E). The model, centers, branch counts, and normalization ranks are unchanged, and the defaults d_h are constant by the choice of the tail. At a fixed surface the two raw sums are finite, and therefore

Let \Sigma be the finite union of the supports of the two functions q_Y and q'_Y on surfaces. Outside \Sigma the displayed sum is zero. Every summand is nonnegative, so no exceptional place with such a surface center can lose weight. At each surface of \Sigma only finitely many places have positive old or new weight, and there are only finitely many positive weights at centers of codimension at least three. This proves that \mathcal C is finite. Summing the displayed identity over \Sigma, adding the finite comparison in codimension at least three, and canceling the unchanged rank term gives (6.12).

For a small step, Lemma 3.4 says that exceptionality of a global place is the same on the two sides, that the two center containment conditions in the statement are equivalent, and that w(E)\geq w^+(E). If a center is not contained in the exceptional locus, its general point corresponds on the common isomorphism open and its discrepancy is unchanged. The set \mathcal P is finite: each exceptional locus has dimension at most two and has finitely many surface components; a surface center contained in it is one of these components. Proposition 6.1 gives finiteness above each such surface and finiteness of all positive-weight places with lower-dimensional centers, on both sides.

Surface centers not contained in L correspond by proper strict transform to those not contained in L^+. At their general points the places, discrepancies, and normalization primes over the surfaces correspond, so the two local brackets agree. The analogous correspondence holds for the raw terms at lower-dimensional centers outside the exceptional loci. In the exceptional loci, every contributing place is counted exactly once in the raw part on each side of (6.7). This remains true if its center changes from a surface to a curve or point, or conversely. The finite comparison of those raw parts is therefore \sum_{E\in\mathcal P}(w(E)-w^+(E)). The subtracted defaults on the exceptional surfaces have difference -\sum_h d_h(e_h-e_h^+). Hence

where the last equality is Lemma 6.3. This is (6.13). Every summand is nonnegative. A strictly decreasing exceptional weight must have its center in the exceptional loci, since weights agree off them, and is in \mathcal P. A positive weight that becomes zero is included by the definition of \mathcal P. The same positivity is immediate in (6.12); this proves strictness in both cases.□

Proof. By Proposition 4.2, the model X_{i+1} is smooth near the general point of V, the junction morphism is an isomorphism there, and the generic blowup of V gives a unique global place E with

This place is exceptional over X_{i+1} and does not belong to the stabilized extracted set \mathcal I. The blowup of the coherent ideal of V is projective, so Lemma 3.1 gives a smooth common model W carrying both E and the fixed nef datum, with a projective morphism r:W\to X_{i+1}. By Lemma 3.2,

At the generic smooth codimension-two center, the ordinary log discrepancy of the blowup is two. The discrepancy formula is therefore

Both subtracted terms are nonnegative. Thus a^+\leq2.

We verify the denominator in this calculation. By the choice in Section 6, N clears the coefficients of the initial boundary and the Cartier denominator of the higher nef divisor of this strong gklt sequence. Since the downstairs boundaries are strict transforms, NB_{i+1} is an integral Weil divisor. Also NM_{i+1} is an integral Weil divisor: it is the pushforward of the integral Cartier divisor NM_W from a common carrier. On the smooth open neighborhood of the general point of V, these two integral Weil divisors are Cartier. Their orders at E are consequently integral after multiplication by N. In particular,

Only Cartierness on this smooth open has been used. Equation (7.1) gives a^+\in(1,2]\cap N^{-1}\Z.

Put a^-=a_i(E). The strict part of Lemma 3.4 applies because \cent_{X_{i+1}}(E)=V\subset L_i^+, and gives a^-<a^+. The same lemma says that E is exceptional over both downstairs models. Since it does not belong to \mathcal I, it is not an extracted prime on either junction; it is therefore exceptional over both. All the intervening steps are small, so it stays exceptional on every model of the connector.

Crepancy identifies the discrepancies at the junctions with a^- and a^+. Write k=N(2-a^+), a nonnegative integer. Since a^-<a^+\leq2, the ceiling weights satisfy

This includes a^+=2, when k=0. No denominator assertion for a^- is needed. The connector consists of one coefficient change and a finite, possibly empty, string of small steps. Its weights are nonincreasing, so the strict decrease between its ends occurs in at least one move. The place E is exceptional throughout that move. Proposition 6.4 makes the difficulty decrease strictly there and nonincrease in every other move, proving the claim.□

Proof. Suppose such a sequence were infinite. Apply Proposition 4.1, then Proposition 4.2, and use the further truncation and denominator choice of Section 6. At every junction the difficulty is a nonnegative integer by Proposition 6.2. It is nonincreasing from one junction to the next by Proposition 6.4. By Lemma 7.1, it drops by at least one whenever the positive exceptional locus contains a surface. There can be only finitely many such indices. This argument uses the lower bound only at junctions; it requires no lower bound on the intermediate models.

Pass to the remaining tail. There \dim L_i^+<2. The dimension inequality in Lemma 3.4 forces \dim L_i=2, so L_i contains an irreducible compact analytic surface. By Lemma 3.7,

at every step of this tail. These are nonnegative integers, a contradiction.□

Proof. An elementary program extracts no prime divisors. Lemma 3.7 with k=3 therefore permits only finitely many divisorial steps. If the sequence were infinite, we could discard a finite prefix containing them and work on a tail of small steps. The rational b-line realization described in Section 2 has the same discrepancies and fixed higher nef data; common projective carriers for finite portions carry their analytically nef pullbacks. Thus Theorem 2.6 applies. After a further tail, both exceptional loci of every step avoid every generalized log canonical center on their respective sides.

Set H_i=\lfloor B_i\rfloor. Since the pair is glc and B_i is effective, its coefficients belong to [0,1], so H_i is the reduced sum of its coefficient-one components. Each such component is itself a generalized log canonical center. Hence \Supp H_i is disjoint from the exceptional loci adjacent to T_i on this tail. The divisors H_i are effective and \Q-Cartier on the strong models, and they are strict transforms of each other across the small steps.

Remove the whole floor and write

The new boundary is effective. For every global place E, comparison of the two crepant equations gives

The added order is nonnegative. If the old discrepancy is zero, the gdlt definition says that the general point of its center is a stratum of the coefficient-one boundary. Taking its closure places the entire center in \Supp H_i. The order of an effective \Q-Cartier divisor at a place whose center lies in its support is strictly positive. Thus every former zero discrepancy becomes positive, and every former positive discrepancy stays positive. The pairs (T_i,\widehat B_i+\M) are gklt.

We check the two ample signs after this full deletion. For a step write

Lemma 3.4 gives a common exceptional image A with L=f^{-1}(A), L^+=(f^+)^{-1}(A), and both maps isomorphisms off A. Properness makes f(\Supp H_i) closed. It is disjoint from A, because \Supp H_i\cap L=\varnothing. On the base open Z\setminus f(\Supp H_i), which contains A, the divisor H_i vanishes, so -\widehat D_i=-D_i+H_i has the original relative ample sign. On Z\setminus A the map f is an isomorphism, and every line bundle is relatively ample for an isomorphism. These two opens cover Z. After clearing a common Cartier multiple, relative ampleness is local on the base, so -\widehat D_i is f-ample. The identical argument with f^+ and H_{i+1} proves that \widehat D_{i+1}=D_{i+1}-H_{i+1} is f^+-ample.

Reindex this floor-deleted tail at its first model and regard it as a sequence of the arbitrary small diagrams in Theorem 7.2. The model and diagram hypotheses are inherited; the new boundaries are effective and related by strict transform, the actual adjoints are \Q-Cartier, and the two ample signs have just been checked. A common carrier for the finite preceding portion, projective over the new first model, supplies the fixed analytically nef divisor. In this application the detector integer is chosen afresh for the new initial boundary and nef carrier. The theorem makes the tail finite, contradicting its construction.□

Proof. The composite Y\to Z is projective bimeromorphic to a normal compact Kähler base. On a projective common carrier, the fixed analytically nef divisor is nef over Z. Thus the continuation assertion of Proposition 2.4 applies to this gdlt pair and to every non-nef finite prefix produced from it.

The initial adjoint D_Y=h^*D_X is not nef over Z. Indeed, the nonisomorphic projective contraction f has a positive-dimensional fiber, and projective hyperplane cuts give a compact irreducible curve C in that fiber. The negative ample sign gives D_X\cdot C<0. There is a compact irreducible curve C'\subset Y mapping onto C. For completeness, normalize C, take an irreducible component of its base change under h which dominates that normalization, and write r for its generic fiber dimension. The normalization of C is a projective compact curve, so this component, being projective over it, is projective. A sufficiently high very ample system on the component has r general members whose intersection with a generic fiber is a nonempty zero-dimensional cycle. Their intersection therefore has a compact curve component dominating the base. Its image in Y is such a C'. The induced map of normalized curves has a positive finite degree. The projection formula now gives

In particular, any run reaching nefness must contain at least one step.

Choose an elementary negative step whenever the current adjoint is not nef over Z. The relative continuation assertion keeps all outputs projective over Z, strong normal compact Kähler, and gdlt with the transported data. All models in the run are irreducible: Y is bimeromorphic to the irreducible X, and every elementary step is bimeromorphic. If this process never reached a nef adjoint, it would give an infinite elementary gdlt sequence with the fixed analytically nef b-divisor. This contradicts Theorem 7.3. We obtain a finite run of length m\geq1 with end adjoint D_{Y_m} nef over Z.

Take a smooth common model W with projective maps to the models in the run, to X^+, and to the nef carrier, all over Z. Write P_0,P,P_+ for the pullbacks of D_Y,D_{Y_m},D_{X^+}, respectively. By Lemma 3.5, the finite run gives

Initial crepancy identifies P_0 with the pullback of D_X. Lemma 3.4, pulled to W, gives the second comparison

The first end is nef over Z, and D_{X^+} is ample over Z. Lemma 3.6 therefore gives an actual equality P=P_+ and a projective morphism h^+:Y_m\to X^+ over Z with D_{Y_m}=(h^+)^*D_{X^+}.

The end is strong, gdlt, and has effective boundary by the relative program. It remains to check its exceptional primes. Every prime on Y_m has a strict-transform prime on Y, because an elementary program extracts no prime divisors. Let R be exceptional for h^+, and let R_0 be its prime on Y. If R_0 were not exceptional for h, it would represent a prime on X. Smallness downstairs identifies that prime with a prime on X^+. The place of R would then have a divisorial center on X^+, contradicting its exceptionality. Hence R_0 is h-exceptional and has coefficient one in \Theta. Boundary coefficients of surviving primes are unchanged by an elementary program, so R has coefficient one in \Theta_m. The displayed crepant equality also identifies its discrepancy over X^+ as zero. This proves the lemma.□

Proof of Theorem 1.1. Assume that the sequence in the theorem is infinite. By Lemma 3.4, every global discrepancy is nondecreasing along it. Hence every (X_i,B_i+\M) is glc. The rational b-line realization of the initial data satisfies Proposition 2.5: X_0 is a normal compact Kähler fourfold, the boundary is effective and rational, the nef datum has a projective carrier, and the adjoint is rationally invertible. The proposition gives a projective crepant gdlt modification

with a strong normal compact Kähler source and only coefficient-one exceptional primes. It requires no strong factoriality of X_0.

Apply Lemma 7.4 to the first downstairs diagram and h_0. Its endpoint is a crepant gdlt modification of the same type over X_1, so it is the input for the next application. At every finite stage a common projective carrier supplies the fixed analytically nef data over the current downstairs model. Induction therefore lifts every downstairs diagram. Each lifted string is finite and contains at least one elementary step, and its endpoint is used as the literal starting model of the next string. Their concatenation is an infinite sequence of elementary gdlt steps on strong normal irreducible compact Kähler fourfolds, with the same b-divisor throughout. Theorem 7.3 allows the bases to vary and rules out this sequence. The contradiction proves Theorem 1.1.□