Abundance after nonvanishing for compact Kähler fourfolds

Proof. The strict boundary coefficients belong to [0,1], so positive coefficients of H occur only on exceptional primes. Coefficient by coefficient,

The fractional boundary \{G\}=G-\floor G is effective SNC with coefficients below one. The displayed adjoint difference is relatively nef and big for the bimeromorphic p: it is relatively trivial, and the generic relative dimension is zero. Analytic relative Kawamata–Viehweg vanishing [12, Theorem 2.41] gives R^ip_*\OO_U(H-R)=0 for i>0. Exceptional Hartogs extension gives p_*\OO_U(H)=\OO_T. The sequence for the Cartier divisor R therefore yields a surjection

It factors through p_*\OO_R. Multiplication by the canonical section of H embeds \OO_R into \OO_R(H), because R is reduced SNC and shares no component with H. The surjection consequently makes that embedding an isomorphism after pushforward and makes \OO_T\to p_*\OO_R surjective. Its kernel is exactly the reduced ideal of the proper image p(R): a holomorphic function vanishes after restriction to R precisely when it vanishes on that image. This proves (2.4); Stein factorization gives connected fibers.

For the last assertion, take a klt resolution, so R=0. The same vanishing and Hartogs calculation identify Rp_*\OO_U(H)\simeq\OO_T. The factorization

is the identity, as can be checked on degree zero. It splits the first map. Apply proper coherent duality. Canonical higher direct images of the resolution vanish: the canonical torsion-freeness theorem [21, Theorem 2.9 and Proposition 2.11] applies locally on the base, and these higher images vanish generically for a bimeromorphic map. The required local Kählerness follows by restricting to a small Stein base neighborhood and combining a relative ample metric with a pulled-back strictly plurisubharmonic potential. Thus the dual split surjection is the trace

where \omega_T^\bullet is the dualizing complex. It follows that this complex is concentrated in degree -\dim U. On that cohomology sheaf the trace is also injective: its source is torsion-free and the map is generically an isomorphism. Hence it is an isomorphism. Biduality gives Rp_*\OO_U=\OO_T. This is rational singularity, and the concentration of the dualizing complex is the Cohen–Macaulay property.□

Proof. The assertion is the identity map if C is empty. Otherwise give C its reduced structure and write C_1,\ldots,C_s for its irreducible components.

We will construct a positive line on a resolution and use it to give the conormal of the full analytic inverse image a positive presentation. Grauert’s contraction criterion will then apply to that conormal.

An exceptional divisor and one actual positive line

Choose a projective resolution \pi:\widehat X\to X with smooth source which is an isomorphism over X_{\rm reg} and has pure divisorial exceptional locus [12, Theorem 2.13]. Compactness makes the resolution a finite composition of blowups. Its source is compact Kähler: metrics of a relatively ample line can be patched using a partition of unity from the base, preserving positivity on the vertical tangent spaces, and a sufficiently large pulled-back Kähler form makes the curvature positive in every direction. Compactness supplies one constant for this last step.

Put \beta=\pi^*\alpha. Pullback of the metric nef approximations shows that \beta is nef. We check the positive top intersection needed on the smooth source. By bigness choose a Kähler current T_X\in\alpha with T_X\geq\omega_X, where \omega_X is a Kähler form on X. Pulling back its local plurisubharmonic potentials gives

The pulled potentials are not identically minus infinity, since \widehat X dominates X. The form \eta is smooth and semipositive. Fix a Kähler form \kappa_{\widehat X}, and for \delta>0 choose a positive form b_\delta\in\beta+\delta[\kappa_{\widehat X}]. Wedge the positive current T-\eta, successively, with b_\delta^2, \eta\wedge b_\delta, and \eta^2. Integration and \delta\downarrow0 give

The last inequality holds because \eta is positive on a nonempty open set. All these products are on the smooth compact \widehat X; only a current wedged with smooth semipositive forms was used.

For an irreducible d-dimensional positive-dimensional subspace V\subset\widehat X, projection of its fundamental cycle gives

The second case maps a null subspace into C. In the first case its image is contained in the image of the exceptional locus, hence in \operatorname{Sing}X. Normality of the threefold makes that singular locus at most one-dimensional. Thus every null subspace of \beta maps into the set C\cup\operatorname{Sing}X of dimension at most one.

The smooth nef positive-volume criterion [12, Lemma 2.35] makes \beta big by (3.2). Collins–Tosatti [8, Theorem 1.1] identifies its non-Kähler locus on \widehat X with \operatorname{Null}(\beta). Choose the Kähler current S\in\beta supplied by [3, Section 2.5.1 and Theorem 3.17(ii)]. In that convention its analytic singularities are defined by one coherent ideal \mathfrak a on \widehat X and one coefficient c>0, with E_+(S)=V(\mathfrak a)=\operatorname{Null}(\beta) as sets. Principalize this global ideal by a finite sequence g:Y\to\widehat X of blowups with smooth centers, using [12, Remark 2.14]. The source Y is smooth and compact Kähler. If \mathfrak a\OO_Y=\OO_Y(-D_{\mathfrak a}), the logarithmic presentation and Poincaré–Lelong, in the cited normalization, give

where G is a finite real divisor supported over the singular set and \theta is a global smooth closed form. If S\geq\epsilon\kappa_{\widehat X}, the equality off G and continuity show that \theta\geq\epsilon g^*\kappa_{\widehat X} everywhere.

Discarding blowups along Cartier centers, which are isomorphisms, choose for this finite composition an effective g-exceptional integral divisor F with \OO_Y(-F) g-ample. Such a divisor is obtained from the exceptional tautological divisors, giving earlier pullbacks sufficiently large positive weights. A smooth representative v of c_1(F) can be chosen so that g^*\kappa_{\widehat X}-\delta v is Kähler for some \delta>0: the curvature of -F is positive on the vertical kernels, and compactness bounds the horizontal and mixed terms. If g is an isomorphism, take F=0. Consequently \theta-\epsilon\delta v is Kähler. With p=\pi g, we obtain

Every prime of G maps into C\cup\operatorname{Sing}X, and every prime of F maps to a set of codimension at least two in \widehat X. Hence every prime of E is p-exceptional. The morphism p is projective: projective morphisms compose over the compact base X by [12, Remark 2.11].

Openness of the Kähler cone on the smooth Y permits a nonnegative rational divisor E' with the same exceptional prime support and coefficients sufficiently close to those of E so that

Choose r>0 clearing its denominators, and put

Thus H is an integral exceptional Cartier divisor and L is an actual holomorphic invertible sheaf.

We verify relative ampleness on every full fiber of p. Let a be a smooth representative of \alpha with local potentials on X, let \Theta_L be the curvature of a smooth metric on L, and choose a Kähler representative for \kappa'. Equality of Bott–Chern classes on Y gives a global smooth function \varphi with

Adjust the metric of L by \varphi. On a base chart a=dd^c\rho, adding r\rho\circ p to its local weights makes their curvature r\kappa'. On a fiber, that added function is constant. The restricted weights therefore have strictly plurisubharmonic ambient extensions. The same weights define positivity on the full possibly nonreduced fiber: nilpotents do not change their values or the absolute values of transition units. The compact analytic positivity criterion and the proper fiberwise criterion [22, Definition 2.8, Corollary 1.12, and Remark 3.2] now show that L is p-ample.

Numerical ampleness on the projective inverse image

The compact reduced curve C is projective, even if it is reducible or disconnected. Indeed, choose one point on each irreducible component which is smooth on all of C and is outside the other components. Their sum is an effective Cartier divisor. Its canonical section is nonzero and has a zero on each positive-dimensional irreducible subspace of C. The positivity lemma in Section 3.4 of [26], followed by Section 3.2, Satz 2, makes its line positive and ample. Fix a very ample line A on C.

Put

The reduced base change of p is projective over C. Since C is compact and projective and the final base is a point, the compact-base projective composition assertion makes D projective [12, Remark 2.11]. Chow and GAGA allow D and its holomorphic invertible sheaves to be treated as a projective scheme over \C [50]. The space D can be reducible, nonnormal, and of mixed dimension. The restriction L_D is the restriction of the actual line L; a component of D can lie inside H.

Choose an ample line A_D on D with a smooth curvature form \Theta_{A_D}. For one integer q>0 sufficiently large,

is positive. To see this on the singular space, extend the local weights of A_D to finitely many relatively compact ambient charts covering D, and bound their Hessians by the ambient Kähler form. This is the local-potential positivity convention. If \Gamma\subset D is an integral curve, its image is a point or one of the C_l; projection on the normalizations gives (p^*\alpha)\cdot\Gamma=0. Therefore

The class of L_D^q\otimes A_D^{-1} is consequently in the dual of the closed cone of curves of the projective scheme D. The numerical space is finite-dimensional. Kleiman’s line-bundle criterion [20, Section 4.8 and Theorem 4.10] says that the ample A_D is positive on every nonzero class of the closed curve cone. Its minimum on a compact unit slice is positive, so its class is in the interior of the dual cone. The sum of an element of the dual cone and an element of its interior is again in the interior. Thus L_D^q, and hence L_D, is ample. The ample subtraction in (3.6) supplies the uniform margin; strict positivity on individual curves alone would not give this conclusion.

The ample line L_D is the positivity needed for the contraction. The remaining ideal calculations transfer it to a positive presentation of the conormal of the full analytic inverse image, including its nonreduced structure.

The full scheme inverse image

Let \mathfrak c=\mathcal I_C, and define

The space Z has its full analytic subspace structure, including possible multiplicities and embedded components, and Z_{\red}=D. Put K_m=p_*L^m for m\geq1. It is a coherent ideal of \OO_X: effectivity of H gives L^m\subset\OO_Y, and p_*\OO_Y=\OO_X by normality and proper bimeromorphicity.

We use relative Serre vanishing in the following exact form. For a proper holomorphic map, a relatively ample invertible sheaf, a fixed coherent twist, and a compact part of the base, its positive higher direct images vanish there for every sufficiently large tensor power. This is [2, Definitions 2.1–2.2 and Proposition 2.5]; it permits nonreduced complex spaces and assumes they are separated and countable at infinity. Those conditions hold for the present spaces and the relatively compact opens used next.

On a base open U choose generators g_1,\ldots,g_t of \mathfrak c. Let Q_U be the coherent kernel in

Choose finitely many such opens with compact subsets whose interiors cover C. Relative Serre vanishing for Q_U\otimes L^m on these compact subsets and for J\otimes L^m over C gives one integer M for which the needed R^1p_*’s vanish for every m\geq M. Tensor the displayed presentation by L^m and push forward. Inside K_m, the image of K_m^{\oplus t} is exactly multiplication by the g_i. Hence, at every stalk over C,

This is an image computation; it uses no projection formula for the possibly nonflat ideal \mathfrak c.

Push forward the exact sequence

The other R^1 vanishing and (3.8) give the canonical isomorphism of coherent \OO_C-modules

In particular the full inverse Z, rather than only its reduction, occurs in this equality. No flatness or Cartier assumption on J has been used.

A positive presentation of the conormal

Let \mathfrak n=\ker(\OO_Z\to\OO_D). This coherent nilradical satisfies \mathfrak n^e=0 for one e, by local Noetherianity and compactness of Z. For

the filtration \mathfrak n^jT_m has quotients

Their coefficient sheaves are fixed and coherent on the projective D. Serre vanishing for the ample L_D kills their H^1 for every sufficiently large m. Induction on this finite filtration then gives

Only H^1 of the subobject and quotient is needed at each step, so the embedded nilpotent layers are included.

Write F_m=f_*(L^m|_Z). Projection formula with the invertible A^{-2} and the low-degree Leray sequence give

This does not assert vanishing of R^1f_*T_m. Embed i:C\hookrightarrow\PP^n by A. By GAGA the coherent sheaf F_m is algebraic. The sheaf i_*(F_m\otimes A^{-1}) is 0-regular: its degree-one condition is (3.10), and all higher conditions vanish since its support has dimension at most one. Castelnuovo–Mumford regularity [54, Tag 08A8, Lemma 33.35.12] gives a surjection

Coherent torsion is allowed in this argument.

Fix one such m. We first check that C\subset V(K_m). For each C_l, the projective D contains an integral curve \Gamma dominating C_l: take a component dominating C_l and cut by sufficiently general ample hyperplanes, taking the component itself when it is a curve. From (p^*\alpha)\cdot\Gamma=0 and \kappa'\cdot\Gamma>0 we obtain E'\cdot\Gamma<0. If \Gamma were not contained in the support of the effective Cartier divisor H, its canonical section on the normalization of \Gamma would give nonnegative degree, a contradiction. Thus an irreducible component of H contains \Gamma. Its image contains C_l and has dimension at most one by exceptionality, so that image equals C_l. At every point of C_l a unit germ on X pulls back to a unit and cannot vanish along that component. Hence K_m is a proper ideal at every point of C.

Outside p(H), the ideal K_m equals \OO_X. The image of each exceptional prime has codimension at least two in the normal threefold. It follows that V(K_m)_{\red} is a finite union of curves and points. Since it contains every C_l, each C_l is one of its irreducible curve components. Let B be the union of its irreducible components other than the selected curves C_l, and put \mathfrak b=\mathcal I_B. Define

This is a coherent ideal with one global finite saturation exponent. Indeed, for each integer j\geq0, the ideal (K_m:\mathfrak b^j) is the kernel of the coherent morphism \OO_X\to\mathcal Hom(\mathfrak b^j,\OO_X/K_m). The ascending chain stabilizes at each stalk by Noetherianity. Equality of two successive coherent ideals then holds on a neighborhood, and the colon recursion gives all later equalities there. Compactness supplies one exponent for all of X.

On X\setminus B we have I=K_m. On X\setminus C, the analytic Nullstellensatz puts a local power of \mathfrak b inside K_m, so I=\OO_X. Closure of the dense parts C_l\setminus B now gives

The set B\cap C is finite. The induced map

is therefore an isomorphism away from finitely many points of C. After tensoring by A^{-2}, its kernel and cokernel are still point-supported. Splitting through its image, the two long exact sequences, (3.10), and the vanishing of positive cohomology of point-supported sheaves give H^1(C,Q\otimes A^{-2})=0. Here a coherent sheaf on the projective curve has no H^2. The same regularity argument as above gives

Since I\subset\mathfrak c, right exactness of restriction gives the actual conormal identity

Its support is all of C. At each point there, I is a nonzero proper finitely generated ideal and \mathfrak c lies in the maximal ideal, so Nakayama gives I/\mathfrak cI\ne0. Nonzeroness of the ideal also follows from its one-dimensional zero set in the normal threefold.

The coherent conormal contraction criterion

For a coherent ideal, Grauert’s associated normal linear space is defined by its local linear relations. Restricting a finite presentation of I to C presents the module in (3.13). Thus the reduced associated space in Sections 3.6–3.7 of [26] is

This definition does not require Q to be locally free. The surjection (3.12) embeds N_I as a closed reduced linear subspace of \operatorname{Tot}((A^{-1})^{\oplus N_Q}).

Put V_A=H^0(C,A)^*. The very ample embedding C\hookrightarrow\PP(V_A), where \PP(V_A) parametrizes lines, identifies A^{-1} with the restricted tautological line. Consequently \operatorname{Tot}((A^{-1})^{\oplus N_Q}) is closed in C\times V_A^{N_Q}. Its projection to V_A^{N_Q} is proper because C is compact, and it is an analytic embedding off the zero section: on the open set where its j-th vector is nonzero, projectivization to that vector’s line, followed by the inverse of the embedding of C on its image, recovers the base point holomorphically. The squared Euclidean norm of this projection is strictly plurisubharmonic off the zero section. Its positive sublevels are relatively compact neighborhoods of that section with strongly pseudoconvex boundaries; scalar multiplication gives a nonzero radial derivative at every positive level. Restricting them to N_I proves exactly the weak negativity of this reduced linear space, including when C is singular or disconnected and Q has torsion.

Grauert’s Section 3.7, Satz 8, applies to any coherent ideal with the given compact zero set and with weakly negative associated normal linear space [26]. It does not require the ideal to be radical or locally principal. The ambient X is reduced and normal, and its compact zero set C is a union of curves with no isolated points. The criterion therefore makes C exceptional. Section 2, Definition 3 and Satz 5, of that source give a global proper surjective holomorphic map \phi:X\to X' with \phi(C) discrete, \phi^{-1}(\phi(C))=C as underlying sets, and

The Remmert reduction used there has connected fibers. Hence each connected component of C has one image point and two components cannot have the same image. Its normality statement for a normal source (p. 337) makes the Remmert-reduced target neighborhood normal; gluing it to the unchanged normal complement makes X' normal. Compactness of X makes X' compact, and the complement isomorphism makes \phi bimeromorphic. This proves the proposition.□

Proof. Use a projective resolution \pi:\widehat X\to X with smooth compact Kähler source as above. The class \beta=\pi^*\alpha is nef with positive cube by (3.2). Collins–Tosatti makes \operatorname{Null}(\beta) a proper analytic set.

Each irreducible component Z of this analytic set is itself null. It has positive dimension, since each of its points lies on a positive-dimensional null subspace. Suppose instead that \beta^{\dim Z}\cdot Z>0. Resolve Z projectively. The restricted pulled class on the smooth compact Kähler resolution is nef with positive top intersection, so Collins–Tosatti gives it a proper null set. Choose a point of Z outside the image of that set, the target nonisomorphism locus of the resolution, the singular locus of Z, and the other components of \operatorname{Null}(\beta). Such a point exists because these are proper analytic subsets of Z. By the definition of \operatorname{Null}(\beta), a null irreducible subspace passes through it. That subspace lies in Z, since the point is on no other component. Its strict transform is null for the restricted class, contradicting the choice of the point.

There are finitely many components of the compact analytic null set. By (3.3), each of its surface components is exceptional, since a nonexceptional one would map to a null surface on X. Thus their images, as well as those of its curve components, have dimension at most one. For any downstairs null curve C_0, choose an irreducible component V of \pi^{-1}C_0 dominating C_0. Proper surjectivity supplies such a component. If \dim V=1, cycle projection gives \beta\cdot V=\deg(V/C_0)\,\alpha\cdot C_0=0; if \dim V>1, it gives zero by dimension drop. Thus V lies in an upstairs null component whose image contains C_0. Every downstairs null curve is therefore contained in the finite union of images. An irreducible curve in a finite union of curves and points is one of the curve components. There are therefore only finitely many downstairs null curves, as asserted.□

Proof. Choose an integer N>0 clearing the actual adjoint index and the Cartier indices of S+B. On the common codimension-one locally free open, and hence everywhere by reflexive extension,

Thus this reflexive canonical power is an actual line. In particular K_X+S is an actual rational adjoint; no global canonical Weil generator is being chosen. Subtracting the effective \Q-Cartier boundary from discrepancies shows that (X,S) is plt and X is klt. Ordinary plt adjunction [9, Lemma 5.3] gives normal S and klt (S,B_S). With compatible residue embeddings, restriction of the canonical section of a Cartier multiple of B gives

The coefficient computation below also proves \Theta\geq0. The notation S|_S always denotes the restriction of the actual rational normal line.

The analytic quotient in codimension two

Fix a prime P on S. Near a general point choose a holomorphic equation v for a Cartier multiple mS, and normalize the full finite root cover

The root algebra is reduced before normalization. It is free with basis 1,t,\ldots,t^{m-1} over the normal local domain. Its generic Kummer algebra is reduced in characteristic zero because v\ne0, and freeness makes the map to that generic algebra injective. A nilpotent element must therefore be zero. Finite analytic normalization [34, Part B, Section 4, Corollaries 2–3] gives a finite normal cover \tau:Y\to U with its complete \mu_m-action. It can be disconnected, and we retain all components. The invariant subalgebra of its normalization is \OO_U: in the total meromorphic algebra the invariants are the meromorphic field of U, and normality of U identifies its integral elements. Thus the quotient is the original normal space.

This cover is étale at every codimension-one point. Off S, the equation is a root of a unit. At a general smooth point of S, write v=u x^m for a unit u; after choosing a holomorphic root of u, normalization is a disjoint union of the branches t=\zeta u^{1/m}x. The lifted floor

is reduced and Cartier. Its prime coefficients are one, and on a normal space the principal divisorial ideal with these coefficients is the radical ideal of their union.

The finite discrepancy formula is valid here in the analytic category. Normalize a base change of a model carrying a tested divisorial valuation. The Hurwitz formula for the corresponding discrete valuation rings and the actual identity K_Y+S_Y=\tau^*(K_X+S) give

Extensions of the valuation of S are exactly the height-one valuations of the lifted floor. All exceptional log discrepancies remain positive, so the lifted pair is plt. Applying the same formula without the floor shows that the components of Y are klt.

The lifted floor is normal. One can use the ordinary plt adjunction just cited, or the following independent connectedness argument. On a projective SNC resolution of the lifted pair, its coefficient-one locus is the union of the strict floor components, with no exceptional component. Two such components cannot meet: blowing up their intersection would give an exceptional log discrepancy zero. They are therefore disjoint and smooth. Lemma 2.1 gives the equality of their direct-image structure sheaf with that of the reduced floor. Factoring the map through the finite normalization of the floor, and applying normal bimeromorphic Hartogs extension on that normalization, identifies this direct image with the normalization sheaf. The equality forces the normalization to be an isomorphism. This use of the connectedness lemma is independent of any contraction existence assertion.

Choose a component P' above P and a general point on it. The normal S_Y is smooth there, since P' has codimension one in it, and P' is smooth after a further generic choice. The ambient Y is smooth there as well. Indeed, in its local ring R' the nonzerodivisor t has regular quotient R'/(t). Lifting its minimal generators and adjoining t bounds the embedding dimension of R' by \dim R', so R' is regular.

Remove the finitely many proper fixed loci on P'. At a remaining point its stabilizer G\subset\mu_m fixes P' pointwise. Holomorphic linearization can preserve the smooth flag P'\subset S_Y\subset Y: average lifts of an eigenbasis of the cotangent space inside the respective invariant ideals, and use their independent differentials as local coordinates. The coordinates tangent to P' are fixed. The stabilizer has no ineffective element: its action sends t to \zeta t, and the nonzerodivisor t forces \zeta=1 for an element acting trivially on the local germ. A nonidentity stabilizer element cannot fix a transverse hyperplane, because \tau is étale in codimension one. Both transverse characters are consequently faithful. If r=|G|, rescaling the generator gives the analytic germ

The smooth case is r=1. The local Cartier index of S is r: x^r descends, whereas invariance of a unit times x^d, evaluated at the fixed point, forces r\mid d.

On the quotient floor let z=y^r. Residue of the r-th log-canonical power changes (dy)^{\otimes r} into a nonzero constant times z^{-(r-1)}(dz)^{\otimes r}; the tangential factors are unchanged. The actual meromorphic residue embedding therefore gives

This derivation is on the analytic cover and does not infer a higher-dimensional analytic quotient theorem from an algebraic slice.

Depth and the canonical multiplication

The sheaves \mathscr I_k are maximal Cohen–Macaulay on X. Here is the index-cover argument at every point of X. On a small open about an arbitrary point, repeat the full normalized root construction using a local equation for a Cartier multiple mS. Codimension-one étaleness and the discrepancy calculation above do not require the generic choice on P. The klt components of this full cover Y are Cohen–Macaulay by the last assertion of Lemma 2.1; its proof uses relative vanishing, canonical torsion-freeness, and coherent duality. The total meromorphic Kummer algebra has basis 1,t,\ldots,t^{m-1}, even when it splits. Its character-j holomorphic summand on this open is

Indeed regularity in each height-one discrete valuation ring requires the coefficient to have order at least -j along S and at least zero elsewhere. Normality supplies the equality of these sheaves. Up to tensoring by the actual Cartier line of a multiple of mS, these are all the \mathscr I_k. Averaging over \mu_m splits them as modules.

For the depth assertion, let R=\OO_{X,x} have dimension d and choose a system of parameters. Every normalization component dominates the base open: the generic Kummer algebra is a separable product of fields, and the full normalization retains all of those components. Thus each local ring of Y above x has dimension d, and finiteness makes the extended parameter ideal primary for its maximal ideal. Cohen–Macaulayness upstairs makes the same parameter sequence regular. Testing at all upstairs maximal ideals makes it regular on the finite semilocal R-module (\tau_*\OO_Y)_x. It remains regular on each split summand, and hence on \mathscr I_k. No flatness of the finite cover is used.

Divisor orders give \mathscr I_j\mathscr I_\ell\subset\mathscr I_{j+\ell}; also \mathscr I_1 is the reduced ideal of S. Thus \mathscr E_k is a coherent sheaf on S. The depth lemma in

gives depth at least \dim X-1 along S. The quotient has generic rank one on the irreducible S, so its support is all of S and has local dimension \dim X-1. Depth cannot exceed this dimension, and equality follows. Thus it is Cohen–Macaulay on S, has no embedded associated component, and is torsion-free and S_2. Normality of S makes it rank-one reflexive.

In (4.4), exactness of finite-group invariants identifies

where \mathbf u denotes the fixed tangential coordinates. Let j_{k,P} be the unique integer with

Each invariant series factors as x^k y^{j_{k,P}}h(y^r,\mathbf u). Hence this is a free rank-one module at the general point of P, and its r-th multiplication into the degree kr layer is

Relative to the local frame (x^r)^k of the actual line \OO_X(-krS)|_S, its vanishing order is j_{k,P}. By (4.5),

Now choose the global Cartier index q in the statement. Multiplication of the divisorial filtration defines a canonical global map

Replacing one factor by \mathscr I_{k+1} raises its product into \mathscr I_{kq+1}, so the map is well defined. The equality on the right follows from \mathscr I_{kq+1}=\mathscr I_1\otimes\OO_X(-kqS). The image \mathscr J_k is a coherent rank-one subsheaf of the actual line \mathscr A_k. Its reflexive hull is that line tensored with the divisorial ideal of an effective integral Weil divisor. At every codimension-one point the source of \mu_k is free and the map is injective. Thus its reflexive hull is canonically the same \mathscr J_k^{**}.

The local index r divides q, so the order of \mu_k at P is qj_{k,P}/r. Define \Xi_k=\sum_P\xi_{k,P}P. It is finite, either from the coherent image on the compact S or from the bound by \Theta\leq B_S. Taking the reflexive image in (4.7) gives exactly (4.1). The canonical map into the specified target line supplies the global identity and excludes an undetermined line-bundle factor.

Vanishing on a small floor model

We use the projective relative MMP in dimension at most three to obtain a small ordinary \Q-factorial model p:S'\to S. For precision, take a projective log resolution of (S,B_S), principalizing the coherent ideal of its finite boundary support; no individual \Q-Cartier assumption on B_S is required. Give each exceptional prime an effective rational coefficient below one and strictly above its crepant coefficient. For the resulting effective SNC klt boundary C_{\widehat S},

with F exceptional and positive on every exceptional prime. This is the actual meromorphic adjoint identity. The adjoint is relatively pseudo-effective. The morphism is a projective surjection between normal compact analytic spaces, the source is smooth and ordinary \Q-factorial, and its dimension is at most three. Thus [11, Proposition 2.26] applies. On its relative minimal model the transform of F is exceptional and relatively nef, so negativity [55, Lemma 1.3] makes it zero. No step extracts a prime, hence the resulting p:S'\to S is small and projective. The source is ordinary globally \Q-factorial and compact Kähler, and codimension-one comparison gives the actual crepant identity

with the strict boundary and a klt pair. This uses the relative projective MMP as an external input; it is separate from the global contraction construction considered here.

The finite strict transform \Xi'_k is \Q-Cartier by ordinary global \Q-factoriality, as is B'_S. Put B'_k=B'_S-\Xi'_k. It is effective by (4.1). Pullback of the effective \Q-Cartier \Xi'_k lowers the crepant coefficients of the klt pair, so (S',B'_k) is klt.

Work over a connected relatively compact Stein open U\subset W whose closure lies in a chosen larger open. Properness and connected fibers make S_U connected, and normality then makes it irreducible. Relative generation for a large twist by a \phi-ample line [12, Theorem 2.12], followed by Cartan generation on the Stein base, gives a meromorphic section of \mathscr E_k|_{S_U}: take the quotient of a nonzero section of the twist and a nonzero section of the twisting line. The same argument applies to the actual line \mathscr A_k. Write

for the resulting integral Weil divisor G_k and Cartier divisor H_k. Applying the actual identity (4.1) to these meromorphic sections gives a genuine principal relation

Let G'_k be the strict transform on S'_U. A small map has no local exceptional prime either. Strict transform therefore takes H_k to its Cartier pullback and a principal divisor to that of the pulled-back meromorphic function. It follows that

After clearing the global indices of \Xi'_k, the right side is Cartier. Hence the locally chosen integral G'_k is \Q-Cartier. This local assertion follows from the displayed identity; ordinary global \Q-factoriality supplied the index of the globally defined \Xi'_k. The actual adjoint in (4.8), together with the globally \Q-Cartier boundary B'_S, also makes a reflexive canonical power on S' an actual line. The same local generation argument supplies a meromorphic canonical generator if a local Weil representative is needed for the vanishing statement.

Equations (4.8) and (4.9) give

The right side pulls back the defined full adjoint and normal rational lines, and the locally chosen G'_k is \Q-Cartier by (4.9). The line H is \phi-ample. Projective analytic morphisms compose over a neighborhood of a compact subset of the final base [12, Remark 2.11]. Here W is compact, so \phi p is projective over the whole base; equivalently one may use the compact closures in the local argument. Both p and \phi p are proper surjections.

The difference in (4.10) is nef and big for both maps in the convention of [12, Definition 2.40]. Nefness follows by projecting vertical curves: a curve for \phi p maps to a \phi-vertical curve or a point, and the degree for p is zero. For the relative Iitaka condition, clear the index and write \mathscr H=\OO_S(hH) for a \phi-ample line. Normality and proper bimeromorphicity give p_*\OO_{S'}=\OO_S, so projection formula gives

For large m, the first relative complete system is p followed by the relative embedding from \mathscr H^m. Its image has relative dimension \dim S'-\dim W. The second has image the base S, of relative dimension zero, equal to \dim S'-\dim S. These are exactly the two maximal relative Iitaka dimensions. For p, relative bigness is this zero-relative-dimension condition; the pulled line has degree zero on its exceptional curves.

Analytic Kawamata–Viehweg vanishing [12, Theorem 2.41] now applies to the effective rational klt boundary B'_k, the integral \Q-Cartier divisor G'_k, and each of the two proper surjections. It yields, locally over U,

The bijection of prime valuations for the small map, together with normal divisorial extension, identifies

This is an equality of meromorphic sections satisfying the same codimension-one order conditions. Leray gives (4.2) on U, hence everywhere. This proves the proposition.□

Proof.

Canonical rings and their successive kernels.

All direct images defining \mathscr B_k first use the continuous map |\phi|:|S|\to|W|. The equality \mathscr B_1=\OO_W is the canonical equality for the holomorphic map \phi. Apply derived direct image for abelian sheaves to the conormal sequence. Its derived image on an \OO_S-module is the underlying sheaf of the analytic higher direct image. To see this formally, an injective \OO_S-module is flasque: for open sets V\subset U, the extension-by-zero modules satisfy j_{V!}\OO_V\hookrightarrow j_{U!}\OO_U. Adjunction identifies \operatorname{Hom}_{\OO_S}(j_{U!}\OO_U,J) with \Gamma(U,J), so injectivity of J makes the restriction of sections surjective. Flasque abelian sheaves are acyclic for continuous direct image. An injective module resolution therefore computes both the analytic module higher images and their underlying abelian-sheaf higher images. Thus (4.2) gives exact sequences

The displayed quotients are ring maps; exactness here concerns their underlying abelian sheaves. Proper direct-image coherence for the already holomorphic \phi [29, Theorem 1.1, p. 15-01] makes \mathscr F_k a coherent \OO_W-module.

In \OO_{S_{k+1}}, the ideal \mathscr E_k is square-zero and is annihilated by \mathscr I_1: use \mathscr I_k^2\subset\mathscr I_{2k}\subset\mathscr I_{k+1} and \mathscr I_1\mathscr I_k\subset\mathscr I_{k+1}. Consequently the action of \mathscr B_{k+1} on the ideal \mathscr F_k factors through \mathscr B_1=\OO_W. This is the \OO_W-module structure on the square-zero ideal used in the following construction.

The composites of (4.12) are epimorphisms \mathscr B_k\to\OO_W. Their kernels are

by left exactness and \mathscr I_1^k\subset\mathscr I_k. Since W is reduced, \mathscr N_k is exactly the nilradical. Each stalk of \mathscr B_k is local: an element lifting a unit in \OO_{W,w} has an inverse by a finite geometric series in the nilpotent kernel, whereas an element mapping to the maximal ideal is not a unit. Its residue field is \C. Thus W_k is already a locally ringed space, and it remains to construct local analytic presentations for it.

A local analytic presentation from source functions.

Fix k and w\in W. Choose a local closed analytic embedding of a neighborhood of w into an open polydisc. Lift the finitely many germs of its coordinate restrictions through \mathscr B_k\to\OO_W, represent the lifts on a common neighborhood, and shrink the polydisc. This gives a closed embedding i:U\hookrightarrow D\subset\C^d and sections b_1,\ldots,b_d\in\mathscr B_k(U) reducing to the coordinate restrictions. Put Z=S_{k,U}, the open subspace of S_k with underlying set \phi^{-1}(U). By definition, \mathscr B_k(U)=H^0(Z,\OO_Z).

For a possibly nonreduced complex space, a tuple of holomorphic functions defines a holomorphic map to affine space, functorially on structure sheaves [28, Theorem 1.1, p. 10-02]. Apply this theorem to the b_j. Their point values lie in D, so the map factors as

The same tuple theorem identifies its actual analytic restriction to the reduction with i\phi, because the reduced coordinate functions are those of i\phi. In particular |g|=i|\phi|. The restriction \phi^{-1}(U)\to U is proper and i is closed, so g is proper; nilpotents do not change properness of the underlying continuous map.

The holomorphic map g makes \mathscr A=g_*\OO_Z a genuine \OO_D-algebra. Its underlying module is coherent by the proper direct-image theorem [29, Theorem 1.1]. Directly from the underlying maps, as sheaves of \C-algebras,

Reduction gives an epimorphism \mathscr A\to i_*\OO_U. It is \OO_D-linear because the actual analytic restriction g|_S=i\phi identifies the reduced action of every ambient holomorphic function with its restriction to U. The target is the coherent quotient of \OO_D by the ideal of U. Hence its kernel \mathscr N is a coherent \OO_D-module. It is also a nilpotent ideal. Shrink D once more and choose sections n_1,\ldots,n_r generating \mathscr N as a module. They are actual sections of \OO_Z whose reductions vanish. Since \OO_D\to i_*\OO_U is an epimorphism, subtraction of an ambient lift of a reduction gives the stalkwise module equality

No splitting of the reduction map is used here.

Apply the tuple theorem again to the b_j and n_\nu. It gives

If j:U\hookrightarrow P is u\mapsto(i(u),0), then |H|=j|\phi|, since the n_\nu are nilpotent. Thus H is proper. The sheaf \mathscr A'=H_*\OO_Z is a coherent \OO_P-module and the unit is an algebra map

As sheaves of rings \mathscr A'=j_*(\mathscr B_k|_U), so its stalk at j(u) is \mathscr B_{k,u}=\mathscr A_{i(u)}, and its stalk outside j(U) is zero. The stalk identification uses the cofinality of restrictions of ambient neighborhoods among neighborhoods in the closed subspace U.

The map \theta is surjective on every stalk. At j(u), any element of the target has, by (4.13), the form a_0+\sum a_\nu n_\nu with a_\nu\in\OO_{D,i(u)}. The projection of H to D is g, and the extra coordinate t_\nu pulls back to n_\nu. The element is therefore the image of the convergent germ

At all other stalks the target is zero. The kernel \mathscr Q=\ker\theta is a coherent analytic ideal [27, pp. 9-07–9-08]. It defines a closed complex subspace Y=V(\mathscr Q)\subset P by [27, Definition 2.2 and adjacent construction, p. 9-10]. Its support is exactly j(U), since the target stalk there surjects onto the nonzero ring \OO_{U,u}. Under the homeomorphism with U, its structure sheaf is \mathscr B_k|_U, and its reduction is U.

Every germ of \mathscr Q pulls back to zero by the definition of \theta. The closed-subspace factorization criterion [27, pp. 9-04–9-05] therefore factors H holomorphically through Y. Its sheaf map, under the identified structure sheaf, is the canonical evaluation

Indeed every germ of \mathscr B_k|_U lifts through the surjective \theta to an ambient germ, whose pullback is precisely its value as a section on the inverse image. This proves the assertion for every germ, not only the chosen coordinates.

Gluing and closed transition maps.

The ringed space (|W|,\mathscr B_k) was fixed before any of the local choices. The presentations just constructed make each of its restrictions an analytic space by the local definition and adjacent construction in [27, Definitions 2.1–2.2 and adjacent prose]. On overlaps the identifications are the identity on the same sheaf of \C-algebras, and hence are analytic isomorphisms satisfying the cocycle condition [27, Definition 2.4 and Proposition 2.5]. This proves that W_k is a complex space. It retains the Hausdorff topology of W, and its reduction is canonically W. Write \varepsilon_k:|\phi|^{-1}\mathscr B_k\to\OO_{S_k} for the global sheaf evaluation. It is locally the holomorphic factorization above, so it defines \phi_k. Its reduction is \phi, and its underlying map is |\phi|; thus it is proper and surjective. The direct-image equality is the definition of its target structure sheaf, with the canonical unit as the identification.

For the closed transition, use the construction with H at level k+1 and restrict its coordinate functions to the closed subspace S_{k,U}. They give another proper holomorphic map to the same P, with underlying map j|\phi|. Its unit is the composite

The second quotient stays surjective under the closed embedding j, as seen on its stalks. The last sheaf with this ambient action is coherent by proper direct image. The two units therefore have nested coherent ideal kernels, so their analytic subspaces give the required closed inclusion. These local inclusions glue because their ring maps are the fixed quotients. Naturality of the evaluation maps gives

which proves commutativity with the source inclusions. No retraction W_k\to W, Cartesian square, or realization of the infinite tower as one formal completion is asserted or needed.□

Proof. Choose a positive global index \ell such that \ell S is Cartier and \OO_S(-\ell S) is \phi-ample. Put

Here A is the full effective Cartier divisor, including its nilpotents; its local equation on normal X is a nonzerodivisor. Lemma 4.2 makes f a proper surjective holomorphic map of complex spaces and gives the natural equality f_*\OO_A=\OO_{A'}.

Cartier multiplication gives \mathscr I_\ell\mathscr I_j=\mathscr I_{j+\ell}. Hence, for every integer n>0, on the full space A one has

Its finite filtration by \mathscr I_j/\mathscr I_{(n+1)\ell}, for n\ell\leq j\leq(n+1)\ell, has successive quotients \mathscr E_j for n\ell\leq j<(n+1)\ell. All indices are at least one. These filtration terms are \OO_A-modules, because \mathscr I_\ell\mathscr I_j\subset\mathscr I_{(n+1)\ell}.

Let a:S\hookrightarrow A and i:W\hookrightarrow A' be the closed reduction embeddings. The compatibility in Lemma 4.2 gives fa=i\phi. Pushforward by a closed embedding is exact, so the composition identity for derived direct images of abelian sheaves gives

The long exact sequences of the finite filtration in (4.15) therefore give

This uses no vanishing for \mathscr E_0 and keeps the full Cartier thickening throughout.

The actual line \OO_A(-A) restricts to \OO_S(-\ell S). The underlying map of f is \phi, so the reductions of their fibers, as closed subspaces of X, coincide: a reduced closed analytic subspace is determined by its support. Thus \OO_A(-A) is ample on the reduction of every f-fiber. For a line on a compact nonreduced complex space, ampleness on the reduction implies ampleness on the whole space. Indeed the positive metric from [22, Corollary 1.12] has local strictly plurisubharmonic weights in common ambient embeddings, by its Lemma 2.4. The same weights define a metric on the original line, since the pointwise absolute values of transition units are unchanged by nilpotents. The converse positive-metric criterion gives ampleness on the full space. The proper fiberwise criterion [22, Definition 3.1 and Remark 3.2] now makes \OO_A(-A) f-ample.

We apply the analytic blowdown criterion stated in [9, Theorem 4.2, p. 14], attributed there to [18, Theorem 2, p. 495]. For a reduced complex space, an effective Cartier divisor A with its full possibly nonreduced structure, and a proper surjective holomorphic map f:A\to A', this criterion assumes f-ampleness of \OO_A(-A) and

It supplies a blowing down: a complex target Z containing A' as an embedded subspace, a proper surjective holomorphic map F:X\to Z whose restriction to A is f, and an analytic isomorphism of the complements. In its universal form it also supplies the natural equality of sheaves of rings

Here \mathscr S_{X,A,f} consists along A of the germs in \OO_X whose restrictions lie locally in \operatorname{im}(f^{-1}\OO_{A'}\to\OO_A), and is \OO_X away from A. This is a subsheaf of rings; no module coherence of this particular sheaf is used. The hypotheses of the stated criterion follow from (4.16) and the preceding ampleness argument. No flatness or reducedness of A or A' is required.

The embedded A' is closed because it is compact. Its reduction embeds the original W as a closed subspace of Z. The compatible reductions give F|_S=\phi followed by this embedding. Removing a subspace depends only on its support, so the complement isomorphism is X\setminus S\simeq Z\setminus W. It follows that F^{-1}(|W|)=|S|, and the prescribed restriction to A gives the fiber-set equalities in (4.14). Outside W the fibers are single reduced points. These assertions concern underlying fiber sets over W; they require no Cartesian square of the closed thickenings. The fibers are connected because the floor fibers are connected. The target is compact as the image of compact X.

We next prove normality; it is not an extra conclusion assumed from the blowdown criterion. For every open V\subset Z, a section s\in\OO_X(F^{-1}V) restricts to a section on the open space

The equality f_*\OO_A=\OO_{A'} makes this restriction the pullback of a unique section on V\cap A'. Thus every germ of s along A satisfies the defining condition of \mathscr S_{X,A,f}. The reverse inclusion follows from \mathscr S_{X,A,f}\subset\OO_X. Consequently

as sheaves of rings, the second equality by (4.17). This reasoning uses equality of the displayed open subspaces, not of any closed fiber products.

The equality first makes Z reduced. The complement is dense: the inverse image of a nonempty open of Z is a nonempty open of X, and therefore meets X\setminus S. The complement isomorphism thus makes F bimeromorphic. Let \nu:Z^\nu\to Z be the finite analytic normalization. The finite normalization exists by [34, Part B, Section 4, Corollaries 2–3]. The dominant bimeromorphic map from normal X lifts to Z^\nu. Pullback along this lift gives inclusions, injective on the dense common complement,

The normalization is consequently an isomorphism and Z is normal. The natural direct-image equality also gives connected fibers by analytic Stein factorization.

Finally consider the actual global line \OO_X(-\ell S). Its restriction is ample on every reduced F-fiber: this is clear off W, and over W the reduction is the same embedded reduced space as that of the corresponding \phi-fiber. The nonreduced upgrade and the proper fiberwise criterion used above therefore make this one line F-ample. The same proof works for any actual ambient rational line with \phi-ample restriction after clearing its global index. In particular it works for the negative full adjoint in Proposition 4.1. Thus F is projective. A projective resolution with smooth source of the compact Kähler X is again compact Kähler, and its composite with F is bimeromorphic. Hence Z lies in Fujiki’s class \mathcal C.□

Proof. We make the potential in the class equality explicit. Use the normal-space potential description of Boucksom–Guedj [4, Section 4.6.1], specifically Lemma 4.6.1 and the paragraph following Definition 4.6.2. The soft-sheaf descriptions of Bott–Chern cohomology by smooth functions and distributions first give a global real distribution u_Y for which

Locally write T=i\partial\bar\partial v with v plurisubharmonic and p^*\eta=i\partial\bar\partial h with h smooth. Then u_Y+h-v is pluriharmonic as a distribution. By the cited lemma it is a smooth real part of a holomorphic germ. Thus u_Y is locally represented by a quasi-plurisubharmonic function. These representatives agree as distributions on overlaps and hence, after taking their upper-semicontinuous representatives, agree pointwise. They give one global quasi-plurisubharmonic u_Y, which is locally integrable and locally bounded above.

In local ambient embeddings, p^*g is represented by a smooth semipositive form and \kappa by a smooth positive definite form. A finite relatively compact cover of Y therefore gives a constant C>0 with p^*g\leq C\kappa. Consequently

The map p is a modification between normal spaces, and u_Y has the local upper bound needed in the current descent argument. [12, Corollary 2.32 and its proof, p. 18] gives a global quasi-plurisubharmonic u_Z with (4.18) for c=C^{-1}. It is an L^1 potential, and the formula keeps the descended current in the chosen Bott–Chern class. The local potentials of \eta give local potentials for that current. No Kähler form on Z has been used.□

Proof. By [12, Lemma 2.3 and Proposition 2.4], the natural pairing is perfect, \operatorname{Nef}(Z)=C^*, and the Kähler cone is open with closure the nef cone. In particular \kappa lies in the interior of C^*. A sequence in \Sigma with unbounded norm would, after division by its norm and passage to a subsequence, give a nonzero class z\in C with \kappa\cdot z=0, contradicting that interior property. The slice is closed and hence compact.

If it is nonempty, let \mu=\min_\Sigma\ell>0. Homogeneity gives \ell-\frac\mu2\kappa\in C^*=\operatorname{Nef}(Z). A smooth local-potential nef approximation for this last class, bounded below by -\mu\kappa/4, becomes a positive representative after adding \mu\kappa/2. Thus \ell is Kähler. If \Sigma is empty, then C=\{0\} by the interior property; the same conclusion follows because every class is nef and a positive small multiple of \kappa can be subtracted first. All cones here are the local-potential cones in the cited perfect pairing.□

Proof of Proposition 3.1. The proof of [12, Theorem 5.5] first uses a projective relative small \Q-factorialization and then [11, Theorem 1.7]. The latter proof passes to a small strongly \Q-factorial model and treats a nef nonbig class by its Theorem 5.5 and a nef big class by its Theorem 6.4. Here strong \Q-factoriality means that every global coherent reflexive rank-one sheaf has an invertible reflexive power. It implies ordinary global \Q-factoriality; it does not assert the analogous property for all analytic open subsets. The small models use the projective relative MMP of [11, Proposition 2.26 and Lemma 2.27], whose maps are already projective. Compactness of the final bases is retained when projective maps are composed, as required by [12, Remark 2.11].

The imported programs and target descent. The nef nonbig branch follows the chain Theorem 5.5, Corollary 5.4, and Theorem 5.2 of [11]. It uses Step 4 in the proof of [32, Theorem 1.4, p. 242], which contracts a negative-definite collection of curves on a smooth compact Kähler surface and descends through a graph. We use that nonbig argument as an external input. The pseudo-effective cone lineage also uses the terminal canonical threefold program: Assumption 10.1 of [13] invokes its nonvanishing Theorem 9.1, whose proof at p. 50 invokes [32, Theorem 1.1] for a terminal K-MMP and Mori fiber space. We retain that terminal program as an external input. Together with the projective relative MMP just specified, these are the birational inputs used below.

For an already constructed negative-ray contraction with ordinary \Q-factorial compact Kähler dlt source and relatively ample negative adjoint, [6, Proposition 3.1] gives rationality of the target and descent of the supporting class. We use the proof of its Corollary 3.1 for target positivity. The corollary assumes non-uniruledness to obtain a nef support; our ray already has an exposed nef support. Its remaining argument descends that class, proves positivity on the target current cone, and lifts target curves through the projective morphism. These steps apply to the data just stated. The proposition and corollary numbers here are those of the journal edition. We now verify the finite-null and prime-floor constructions to which this argument will be applied.

Nef and big classes with a finite null locus. Three geometric situations use Proposition 3.2: a small negative ray, the entire null locus at a stopping model, and a small ray with pseudo-effective adjoint. The big threefold program uses strongly \Q-factorial klt sources. The last situation can also occur for ordinary dlt data, whose underlying space is ordinary \Q-factorial and klt, as checked below. In every case the relevant locus is the full top-intersection null locus of the proposition.

For a small negative ray, the small branch of [11, proof of Theorem 4.16, p. 40] has, after its boundary perturbation, a strongly \Q-factorial klt source and a nef big supporting class a=(K+\Gamma')+\kappa with \kappa Kähler. At this input [11, Lemma 4.5] says that an a-null surface is covered by a-null curves. All those curves lie in the exposed ray. Such a covering of a surface contradicts the smallness of that ray. Lemma 3.3 therefore makes \operatorname{Null}(a) a finite union of curves, and Proposition 3.2 constructs its contraction. This argument permits a non-pseudo-effective underlying adjoint.

For the entire null locus at the final model X_n of the program in [11, Claim 6.5 and proof of Theorem 6.4, pp. 50–51], the pair is strongly \Q-factorial klt, its class a_n is nef and big, and every a_n-null curve has nonnegative degree for the running adjoint A_n. These are the stopping data from that program. Its null-surface argument, using Lemma 4.5 of the same source, would cover a null surface by a_n-null curves on which the remainder a_n-c_1(A_n) has positive degree. Their A_n-degrees would then be negative, contrary to the stopping data. There is therefore no null surface. Lemma 3.3 again gives finitely many null curves. Apply Proposition 3.2 to their entire union. This use has no single-ray hypothesis and needs only the proper bimeromorphic map to a normal compact analytic target supplied by that proposition.

For a small ray with pseudo-effective adjoint, the branch in the proof of Theorem 6.4 at p. 50 invokes the small case of Theorem 2.23(1) of the same source; this case can also occur in its Theorem 3.1, Claim 3.2. Let A be the negative adjoint and b an exposed nef support for the ray. It may be scaled so that b-c_1(A) is Kähler. To check this usual support step, normalize the closed cone of positive bidimension-(1,1) classes by a Kähler class. Its normalized slice is compact. The continuous function -c_1(A) is strictly positive on its intersection with the exposed ray, hence on a neighborhood of that intersection. On the compact complement, the support has a positive minimum. A large multiple of the support minus c_1(A) is consequently strictly positive on the full slice. Lemma 4.5 makes this difference Kähler. Since A is pseudo-effective, the scaled b is big. The same Lemma 4.5 and the smallness of the ray exclude null surfaces, so Lemma 3.3 and Proposition 3.2 apply. At the outer Theorem 6.4 use the pair is klt. If this step is made for the ordinary dlt data in Claim 3.2, the underlying space is still klt, which is the singularity condition of Proposition 3.2; a small decrease of the boundary preserves negativity when a klt adjoint is needed in a subsequent result.

In each of the two single-ray situations just described, the constructed map contracts exactly that ray. A positive-dimensional reduced fiber is a finite connected union of compact curves, hence is projective. For the corresponding negative adjoint A, a positive global Cartier multiple of the actual line -A has positive degree on every irreducible component of that fiber, and is therefore ample on the fiber. It is ample also on every zero-dimensional fiber. Ampleness passes from a reduction to its nilpotent thickening by the positive-metric criterion [22, Lemma 2.4 and Corollary 1.12]. The proper fiberwise criterion [22, Definition 3.1 and Remark 3.2] now makes this one actual line relatively ample, so the contraction is projective. Normality and proper bimeromorphicity give connected fibers. The source is ordinary \Q-factorial klt, or dlt in the indicated variant, and the exposed nef support is unchanged. Thus [6, Proposition 3.1 and proof of Corollary 3.1], applied as specified above, gives the Kähler target and the descended supporting class. Thus Proposition 3.2 supplies the normal compact analytic contraction, and this postprocessing supplies projectivity and positivity in the single-ray cases. The entire-null cleanup uses the different argument that follows.

In that cleanup, write \mu:X_n\to Z for the entire-null contraction just constructed. The graph argument in [11, proof of Theorem 6.4, pp. 51–52] descends its composite with the program to a morphism \psi:X_0\to Z, where X_0 is the original strongly \Q-factorial klt source of that theorem. All steps of the program are trivial for the transported supporting class. Before applying its Lemma 2.44 to \psi, restore the original boundary \Gamma^{\rm orig} and the original nef big difference \omega^{\rm orig} on X_0, before the internal boundary replacement. The original supporting class is numerically trivial on its contracted curves, and hence

The left side is \psi-nef and is \psi-big for the bimeromorphic map. The rationality result of Lemma 2.44 thus applies to this original Q-Gorenstein klt pair and makes the target Z rational. This step uses the original nef big difference; the transported remainder a_n-c_1(A_n) on the stopping model is not substituted into (4.19).

For the entire-null contraction \mu, the source X_n is a normal compact Kähler klt threefold, so it has rational singularities. The target is normal and compact, and is in Fujiki’s class \mathcal C: a projective resolution of X_n with smooth source is compact Kähler, and its composite with \mu is bimeromorphic. The exceptional image E_Z=\mu(\operatorname{Null}(a_n)) is finite. Every curve contracted by \mu is a_n-null. Apply [11, Lemma 2.11] to this map between normal compact rational spaces in class \mathcal C. It gives a smooth real closed (1,1)-form \eta_Z with local smooth potentials such that

The normalized graph constructed \psi earlier; this application of Lemma 2.11 uses \mu.

We check the three analytic positivity hypotheses on Z. Fix a smooth positive Hermitian form g_Z in local ambient embeddings. The bimeromorphic nef descent theorem [33, Lemma 3.13] applies to the normal compact threefold Z in class \mathcal C and the normal compact threefold X_n. It makes [\eta_Z] analytically nef from (4.20); it does not assume that Z is Kähler. If its defining approximation uses a different positive reference form, compact comparison with g_Z and rescaling the approximation parameter give, for every \delta>0, a smooth function f_\delta with

Changing a smooth representative of the same local-potential class only changes f_\delta by a global smooth potential.

The big class a_n on the compact Kähler source contains a positive closed current T_n with local potentials and T_n\geq\kappa_n for a Kähler form \kappa_n. In view of (4.20), Lemma 4.4 applied to \mu gives a global quasi-plurisubharmonic, hence L^1, function u_Z and a constant c>0 satisfying

This establishes the required current bigness on the not yet Kähler target in the selected class [\eta_Z].

Finally let V\subset Z be an irreducible reduced analytic subspace of dimension d>0. Its strict transform V' is the closure of the inverse image of V\setminus E_Z. It has dimension d, maps bimeromorphically to V, and is not contained in \operatorname{Null}(a_n). Nef approximations on X_n, their restriction to the integration cycle of V', and Stokes’ theorem give a_n^d\cdot V'\geq0. Equality would put V' into the null locus by its definition, so the inequality is strict. The proper cycle identity \mu_*[V']=[V] and projection formula therefore give

This includes V=Z; it requires neither V nor V' to be normal. Additivity over the top-dimensional irreducible components gives the same positivity for every positive-dimensional compact reduced analytic subspace.

Equations (4.21), (4.22), and (4.23) are the three hypotheses of [12, Theorem 2.29]. Apply that theorem directly to \eta_Z. It gives a smooth potential making \eta_Z positive definite, so [\eta_Z] is Kähler and Z is compact Kähler. This verifies the positivity conclusion at the cleanup contraction.

Divisorial contractions. In the point case, let P be the selected prime surface and \nu:P^\nu\to P its normalization. The nef-dimension-zero condition for the supporting class is taken on P^\nu; the whole normalized fiber has zero restricted class by [32, Theorem 3.19(a) and its proof, p. 234]. Use [9, Corollary 4.4], whose proof invokes its Lemma 4.3, at the stated data: a \Q-factorial compact Kähler source, a nef big exposed support, ray curves covering P, and zero restriction on P^\nu. The corollary constructs the proper analytic point contraction and its proof supplies an actual ample line \OO_P(-mP) for some m>0. Thus P is projective, and ampleness persists on the full possibly nonreduced fiber by the positive-metric criterion used above. The full-fiber criterion makes -mP relatively ample.

Every curve of P lifts finitely to its normalization. The zero restricted support therefore gives zero degree on every such curve, so its ambient class lies in the exposed ray. The negative adjoint restricted to P is consequently numerically a fixed positive multiple of the negative normal line. Both have actual rational Cartier multiples. Numerical invariance of ampleness on the projective P and then the full-fiber criterion make the negative adjoint relatively ample. The same [6, Proposition 3.1 and proof of Corollary 3.1] then supplies rationality, class descent, and target positivity for this negative-ray contraction. The point-contraction corollary was used to construct the underlying proper analytic map.

For a divisor-to-curve step, Proposition 4.3 supplies the ordinary extension used both in [11, Theorem 3.1, Claim 3.2] and in its replay in Theorem 4.16 of that source. We verify the two ample-line hypotheses for this use. Write (M,\Delta_M) for the strongly \Q-factorial compact Kähler dlt threefold there, A=K_M+\Delta_M, P for the selected prime floor, and R for its exposed ray. The selected ray satisfies

Adjunction makes P a normal compact Kähler dlt surface and makes A|_P its actual full adjoint. For its inclusion i:P\to M, put

Fix a Kähler class \kappa on M and a nonzero r\in R. For z\in F with \kappa|_P\cdot z=1, positivity of \kappa|_P gives i_*z=r/(\kappa\cdot r). Thus on the full compact normalized vertical face

This uses the cone of positive current classes and includes its limits.

Let \zeta be the nef support exposing R. Its restriction has null cone F. The compact-slice argument just used makes \lambda\zeta|_P-c_1(A|_P) Kähler for sufficiently large \lambda: near F the negative adjoint has the uniform positive bound in (4.24), and away from F the support has a positive minimum. Lemma 4.5 on the rational compact Kähler surface turns this bound into a Kähler class. The ordinary dlt case of the surface theorem [10, Theorem 2.32] therefore gives a projective surjection \gamma:P\to W_P with connected fibers onto a normal compact Kähler space, contracting exactly F, with the support pulled back from a Kähler class on W_P. This use is of the surface theorem.

Applying the same compact-slice estimate after adding a sufficiently large pullback of that target Kähler class makes each of -A|_P and -P|_P relatively Kähler over W_P. They have actual global rational Cartier multiples by the factoriality of M. The proper fiberwise positivity criterion makes both lines \gamma-ample, including when a fiber is the whole surface. Put H=\floor{\Delta_M}-P. For small rational \varepsilon>0, the pair (M,\Delta_M-\varepsilon H) is plt with sole floor P. Indeed on a dlt resolution the pullback of the effective globally \Q-Cartier divisor H is effective. Subtracting it preserves the strict inequality for every exceptional crepant coefficient and lowers all the other strict floor coefficients below one. The negative degree of A-\varepsilon H on R persists for small \varepsilon, so the first relative ample line persists by (4.24); the second is unchanged. These are precisely the hypotheses of Proposition 4.3.

That proposition constructs the extension, whose fiber sets show that the ambient map contracts exactly the original ray and is an isomorphism off P. The negative original adjoint is relatively ample by the same full-fiber criterion. Return at once to \Delta_M and A for the running program. For this projective negative-ray contraction, [6, Proposition 3.1 and proof of Corollary 3.1] applies with the already exposed support, exactly as specified at the start of the proof. It gives rationality, class descent, and Kähler target positivity. Strong factoriality is preserved by [11, Lemma 2.5], and an ordinary projective flip, when needed, is supplied by its Theorem 2.24. Thus the original support and boundary are the ones transported to the next step.

Return to the original threefold. The finite-null and divisorial constructions now supply the indicated steps of the cited big-case program, with their required positivity. Together with the external inputs stated at the start, they give the asserted contraction for the data in (3.1) by the proof of [11, Theorem 1.7]. The proof of [12, Theorem 5.5] descends the contraction from the small model. A pulled-back Kähler class has degree zero on a constant curve and positive degree on a nonconstant curve, as seen on its normalization. This proves the asserted curve criterion and completes Proposition 3.1.□

Proof. Das–Hacon–Păun’s dlt modification theorem [12, Theorem 6.1] applies to a compact Kähler lc fourfold with effective rational boundary and \Q-Cartier adjoint. It gives a projective bimeromorphic morphism

with X' compact Kähler and ordinary \Q-factorial, the pair (X',\Delta') dlt, and the adjoint crepant. The input is klt, so equality of discrepancies makes the output klt as well. The crepant equality is an equality of actual rational lines: pull a local pluriadjoint frame to a common resolution with smooth source as in (2.2), compare the crepant coefficients, and extend the identical meromorphic map across codimension two on the normal X'. Thus for a common index

There is no possible undetected flat-line difference in this comparison.

The pulled-back line is analytically nef by the metric criterion. Also, normality and projection formula identify all its sections with those of \OO_X(mD). The meromorphic maps agree on the common dense open, so their Iitaka dimensions agree. Theorem 4.1 of Höring–Lazić–Lehn [31] states that a nef adjoint of a compact ordinary \Q-factorial Kähler klt pair of positive Iitaka dimension is semiample, unconditionally in dimension at most four. Its analytic positivity and canonical-sheaf conventions are the ones in use here. Apply it to (X',\Delta'), then enlarge the generated degree to a multiple of the original index. All sections are pullbacks. A section nonzero at a chosen point above x\in X descends to a section nonzero at x; Nakayama’s lemma gives the evaluation surjectivity on X.□

Proof. Principalize the reduced coherent ideal of \Supp(\Delta+M), using [12, Theorem 2.13 and Remark 2.14]. This use of an ideal is important: the original boundary need not be \Q-Cartier. The resulting modification has smooth source, is projective, and has locally normal crossing strict and exceptional support. Mark each of its finitely many global prime components separately as an ordered Cartier boundary and apply the ordered-boundary resolution of [53, Theorems 1.1.3 and 1.1.13]. Its final indexed components and their intersections are smooth, and its complete support consists of strict transforms and new exceptional components [53, Lemmas 2.1.10 and 2.2.9(iv)]. Denote the composite by p:Y\to X. It is projective over the compact base. A relative ample metric plus a sufficiently large pullback of a Kähler form makes Y compact Kähler.

Let G_Y be the crepant subboundary for D, defined by the actual meromorphic pluriadjoint pullback. Then p_*G_Y=\Delta, and klt gives \operatorname{coeff}_E(G_Y)<1 at every prime. Define B_Y only after the preceding resolution: assign coefficient one to the actual p-exceptional primes and the strict transforms of primes in M, and retain the coefficients of \Delta on the other strict boundary primes. An exceptional label in the resolution bookkeeping that is not an actual exceptional divisor does not change this rule. This is an effective rational SNC boundary, hence dlt.

Set

Pullback of the holomorphic equation of a Cartier multiple of M makes p^*M effective. On a strict prime of M, the second summand has coefficient 1-\operatorname{coeff}(\Delta)>0; on an exceptional prime it has coefficient 1-\operatorname{coeff}(G_Y)>0; on any other strict boundary prime it is zero. Thus P_Y is effective and its reduced support is exactly the floor of B_Y. The actual identity p^*D\sim_\Q p^*M, together with (2.2), gives K_Y+B_Y\sim_\Q P_Y.

Choose an integer t>0 at least the ratios of the coefficients of P_Y to those of p^*M on every nonexceptional prime of M. The positive part of P_Y-tp^*M is then an effective rational exceptional divisor E, and

For every sufficiently divisible n, exceptional Hartogs extension and projection formula give

The space on the right has dimension one: it is nonzero and \kappa(X,D)=0. The displayed divisor inequality bounds h^0(Y,\OO_Y(nP_Y)) by one, while effectivity makes it nonzero. This controls every sufficiently divisible degree. If two independent sections existed in another Cartier degree, their powers s_0^v and s_0^{v-1}s_1 would remain independent in a degree divisible by the fixed common index, a contradiction. Therefore \kappa(Y,P_Y)=0.□

Proof. Bott–Chern descent first supplies a target class whose pullback is \alpha_i. We prove that this class is nef, contains a current dominating a positive Hermitian form, and has positive top intersection on every positive-dimensional compact reduced subspace. These are the three conditions in [12, Theorem 2.29] that make the target class Kähler.

The descended class and its floor restriction. In this paragraph and the next three, write X=X_i, Z=Z_i, and \alpha=\alpha_i. We use dd^c=\frac{i}{2\pi}\partial\bar\partial, the normalization for which dd^c\phi is the normalized Chern curvature of a metric locally written as e^{-\phi}. A fixed positive rescaling of potentials gives the i\partial\bar\partial convention in the cited analytic inequalities.

Both ends of c are normal compact spaces in class \mathcal C with rational singularities, and \alpha is zero on every contracted curve. The Bott–Chern descent [11, Lemma 2.11] therefore gives a class \beta on Z and a smooth real closed representative \beta_0, with local smooth potentials, such that

Here and below the group is the local-potential Bott–Chern group. Its smooth representative description, including adjustment by a global smooth dd^c-potential, is [33, Definition 3.1 and Remark 3.2].

Let i_W:W\hookrightarrow Z be the embedded floor image and put \delta=i_W^*\beta. The actual identity c|_T=i_W\circ\varphi, (5.3), and (5.7) give

Thus the pullback of \delta-[\omega_W] is zero and hence nef. Apply [12, Lemma 2.38] only to \varphi:T\to W: it is proper and surjective, and both spaces are normal compact Kähler. The lemma makes \delta-[\omega_W] nef. For a smooth representative e of that difference, choose a nef approximation e+dd^c u\geq-\omega_W/2. Then \omega_W+e+dd^c u is a Kähler representative of \delta. In particular its restriction gives a positive current in the class \beta|_V and a positive top integral for every positive-dimensional irreducible V\subset W. Only the nefness of \delta-[\omega_W] enters this deduction.

Nefness on the target. For every positive-dimensional irreducible reduced compact subspace V\subset Z, we first produce a current

with a global real distribution q_V on V. Currents and their positivity on the pure-dimensional reduced V are defined by duality with smooth test forms from local ambient embeddings; see [16, Section 1, Definitions 1.1–1.2, pp. 14–15]. We will use exactly this positive-current meaning of pseudoeffectivity on a possibly nonnormal subspace. The preceding Kähler representative of \delta gives (5.9) for V\subset W, with a smooth potential.

Suppose V\not\subset W, and let V'\subset X be the closure of the inverse image of V\setminus W. The isomorphism X\setminus T\simeq Z\setminus W makes V' irreducible and its map to V proper and bimeromorphic. Choose a projective resolution r:U\to V' with smooth source. The restricted Kähler form makes V' a compact Kähler space. A relatively positive metric for the projective resolution, plus a sufficiently large pullback of this form, makes U a compact Kähler manifold. This metric construction applies to the possibly nonnormal V'.

Let \nu:V^\nu\to V be normalization. The dominant map from the normal U factors through \nu, giving a proper bimeromorphic map p:U\to V^\nu, hence a modification between normal compact spaces. For the other map j:U\to X, functoriality of the actual maps and (5.7) gives

The class on the right is nef. Indeed, pull back the smooth nef approximations from X; on compact U, the pullback of their reference form is bounded above by a fixed multiple of a Kähler form \eta on U, so rescaling the error gives the nef inequalities.

Set \gamma=p^*\nu^*(\beta_0|_V). Choose smooth approximations \gamma+dd^c f_n\geq-n^{-1}\eta. The closed positive currents \gamma+dd^c f_n+n^{-1}\eta have bounded \eta-mass by Stokes’ theorem. Weak compactness gives a positive closed limit in [\gamma]. The \partial\bar\partial-lemma on the compact Kähler manifold U, and the global potential description of a positive current, give a global quasi-plurisubharmonic q_U with

These are also the closed-cone statement following Definition 1.6 and the potential description in (3.1) and its following paragraph in [17, pp. 1253 and 1260]. Apply [12, Corollary 2.32 and its proof, p. 18] to p, with target form \nu^*(\beta_0|_V) and constant zero in its current inequality. The locally integrable, locally bounded above q_U supplies the required potential. We obtain a global quasi-plurisubharmonic q_\nu satisfying

This applies the normal-modification descent to the displayed local-potential inequality on the still general normal compact target.

It remains to push through the finite normalization. Locally write \beta_0|_V=dd^c h with h smooth. The function \nu^*h+q_\nu is locally integrable and locally bounded above, and its Hessian is T_\nu\geq0. On normal V^\nu, its upper regularization is plurisubharmonic. The finite trace is weakly plurisubharmonic and its Hessian is the positive current \nu_*T_\nu by [16, Corollary 1.11 and Proposition 1.13(a), p. 22]. Since \nu has generic degree one, change of variables off the proper analytic exceptional sets gives \nu_*\nu^*(\beta_0|_V)=\beta_0|_V as currents; those sets have measure zero for the smooth top-degree integrands. Proper push of distributions commutes with dd^c. Hence

The traced distributions are locally integrable and glue because q_\nu is global. On a locally reducible V, these local potentials may be only weakly plurisubharmonic; the argument below uses their associated positive currents.

We now apply the sufficient direction of the restriction criterion [12, Theorem 2.36 and Remark 2.37], spelling out its singular-current input. Proposition 3.3(iv), p. 1262, and the following paragraph, pp. 1262–1263, of [17] applies to a compact complex space and a smooth class containing a closed positive current. It gives nefness if the restrictions to the irreducible components of every positive Lelong level set are nef. Induct on the dimension of each irreducible V\subset Z; points are automatic. For a positive-dimensional V, use (5.9). The Siu analyticity assertion accompanying that proposition makes each positive Lelong level set analytic, and it is proper. Indeed, if a fixed positive level filled V, pack disjoint radius-r balls in a coordinate ball of V_{\rm reg}. The Lelong mass lower bound on each ball would force the locally finite mass of the (1,1)-current to grow at least as a positive multiple of r^{-2} when r tends to zero. This is impossible. The level set components therefore have smaller dimension, and their restricted classes are nef by induction. The proposition makes \beta|_V nef. Taking V=Z proves analytic nefness of \beta. This uses the positive-current formulation above also on nonnormal subspaces, where the local potentials may be only weakly plurisubharmonic.

Bigness from the supported section. Use the effective actual representative already present in (5.2). Choose m>0 clearing the indices and an isomorphism of holomorphic lines

The canonical section of the right side gives a nonzero holomorphic section s of L with divisor mP_i. Choose a smooth metric h on L. In a local frame write h=e^{-\phi}, s=f, and put

The first formula defines the normalized global curvature form. The second is a global quasi-plurisubharmonic function, locally m^{-1}(\log|f|^2-\phi). The holomorphic f extends in a local ambient embedding and is not identically zero on any nonempty open subset of that chart in X. Local integrability is Proposition 1.8 of [16, p. 19], and the unnumbered prose after its proof gives the current positivity

The smooth form \vartheta+\omega_i represents \alpha=c^*\beta, so the smooth local-potential representative description supplies a global smooth v with \vartheta+\omega_i=c^*\beta_0+dd^c v. Thus

This is a global quasi-plurisubharmonic potential obtained from the supported section at this running step.

Fix any smooth positive Hermitian form g_Z on Z in local ambient embeddings. Local holomorphic coordinate lifts of c make c^*g_Z semipositive in those embeddings. A finite relatively compact cover of X and comparison with the positive definite ambient representatives of \omega_i give one C>0 with c^*g_Z\leq C\omega_i. Corollary 2.32 of [12] applies to the normal modification c and (5.12). It gives a global quasi-plurisubharmonic, hence L^1, function v_Z with

This is the required dominating current in the selected target class.

Top intersections and the other floor. Let V\subset Z be irreducible of dimension k>0. The case V\subset W was proved using the Kähler class \delta. Otherwise use its strict transform V' above. The proper integration cycle identity c_*[V']=[V], projection formula, and (5.7) give

The right side denotes the integral of any smooth representative of \alpha; Stokes’ theorem makes it independent of that choice. Neither subspace needs to be normal.

If V'\not\subset\Supp P_i, take the projective resolution r:U\to V' with smooth Kähler source used above. The pullback of s is nonzero and defines the effective rational divisor E=m^{-1}\operatorname{div}(r^*s) on U. Put a=r^*(\alpha|_{V'}) and w=r^*[\omega_i|_{V'}]. The class a is nef, w has a semipositive smooth representative, and the actual section gives a-w=c_1(E). Therefore

For each summand, choose a smooth representative of a+\varepsilon[\eta] which is semipositive, where \eta is a fixed Kähler form on U. Restrict it and the semipositive representative of w to every effective divisor component, integrate, and let \varepsilon tend to zero. This proves its nonnegativity, including the degree statement when k=1. Projection formula gives \int_U w^k=\int_{V'_{\rm reg}}\omega_i^k>0. Equations (5.14)–(5.15) give the desired strict positivity.

It remains that V' lies in a component T' of \Supp P_i. It meets X\setminus T, so T'\neq T. The actual floor specialization in Subsection 4.5 gives

where g is projective with connected fibers, its source and target are normal compact Kähler, and \kappa is a Kähler class. The equality is in the local-potential Bott–Chern group. Every irreducible curve in a g-fiber has \alpha-degree zero, so (5.4) puts its class in R_i, and c contracts it.

It follows that h=c|_{T'} is pointwise constant on every full g-fiber. To see this also for nonreduced fibers, their reductions are connected projective varieties, possibly reducible. If the image of such a reduction were not a point, one irreducible component Q would have nonconstant image; otherwise its image would be a connected finite set. At a smooth point of Q where a local coordinate of h has nonzero differential, general hyperplanes through that point cut an irreducible projective curve component whose tangent is not in the differential’s kernel. That curve has nonconstant image, a contradiction. If \dim Q=1, take Q itself.

There is consequently a holomorphic factorization

For clarity, the specialization gives g_*\OO_{T'}=\OO_{\overline W}; this also follows from analytic Stein factorization, connected fibers, and the normality of the two spaces. Proper surjectivity makes g a closed quotient map, so the pointwise constancy first defines a continuous b. For an open O\subset Z embedded in a polydisc, restrict to the open b^{-1}(O). The coordinate functions of h on its full inverse image descend through g_*\OO_{T'}=\OO_{\overline W} to a holomorphic tuple. It lies pointwise in the embedded O; its defining ideal therefore vanishes on the reduced open b^{-1}(O). The resulting local holomorphic maps glue to (5.17). This factorization uses pointwise constancy on the underlying full fibers.

Put H=g(V'), a closed irreducible analytic subspace of \overline W. Since c|_{V'} is bimeromorphic onto V, (5.17) yields

Thus g|_{V'} is generically finite onto H, of some integer degree d>0. The projection formula using (5.16) now gives

The last integral is a positive Kähler volume. The one-way factorization h=b\circ g and this degree calculation give the needed implication in [12, proof of Theorem 7.2, p. 48].

These cases cover every irreducible positive-dimensional V. Additivity over top-dimensional irreducible components gives strict positivity for every positive-dimensional compact reduced subspace. We have proved analytic nefness of \beta, the dominating current (5.13), and all positive top intersections. The normal compact Z and the smooth real closed \beta_0 therefore satisfy the three hypotheses of [12, Theorem 2.29]. That theorem gives a positive smooth representative of \beta. Its local smooth potentials make this a Kähler form \omega_{Z_i}. Equation (5.7) gives (5.6), and Z_i is compact Kähler.□

Proof. Clear the index and pass to the reduction of F. If an irreducible component F_\alpha is not contained in \Supp E but meets it, the restricted canonical section cuts a nonempty effective Cartier divisor on the integral projective F_\alpha. This component has positive dimension: a point that is an irreducible component cannot meet another component, and a connected zero-dimensional fiber is a point. Cutting by sufficiently general hyperplanes of a very ample line on F_\alpha gives a curve on which E has strictly positive degree, a contradiction. If some components are contained in the support and others are not, connectedness gives a noncontained component meeting a contained one and the same contradiction. These alternatives prove the claim, with no reducedness assumption on the original fiber.□

Proof. Write b:\Gamma\to Y for the normal common graph. Choose a relatively very ample and generated line \mathscr L for b; compactness permits one global power. Its evaluation embeds \Gamma in \PP_Y(\mathscr E), where \mathscr E=b_*\mathscr L. This coherent sheaf is torsion-free of rank one: a section killed by a nonzero function vanishes on the dense isomorphism locus and hence everywhere. Since Y is smooth, \mathscr E^{**} is a line, and

is a coherent ideal equal to \OO_Y on U_Y.

Apply the ideal principalization functor of [53, Theorems 1.1.11 and 1.1.13] to \mathscr J. For smooth input it is supported on the cosupport of the ideal, so it avoids U_Y; its total transform is invertible on a smooth space Y_1. The resulting invertible quotient of the pulled-back \mathscr E defines a map Y_1\to\PP_Y(\mathscr E). It lands in the closed subspace \Gamma on a dense open, and hence everywhere because Y_1 is reduced. The map Y_1\to\Gamma is projective: its graph is a closed immersion into the base change of the projective map Y_1\to Y. Its composite to V is projective and is an isomorphism over U_V.

This principalization is not required to have transverse intermediate centers. Instead, after it is complete, mark as separate ordered Cartier boundary components all strict initial boundary primes, all primes in the principalized support, and all divisorial exceptional primes. On U_Y this is the original globally simple SNC list. Apply the ordered-boundary functor of [53, Theorems 1.1.3 and 1.1.13]. Its centers lie over the original labeled bad locus and therefore avoid U_Y. The final indexed components and intersections are smooth, and the total support consists of the strict support and new exceptional support. The already invertible ideal remains invertible, so the map to \Gamma persists.

Finally apply the same two stages to the pullback of the reduced coherent ideal of V\setminus U_V. In the second stage mark, each distinct prime once, every component of the prior resolved support, every component of the newly principalized support, and every divisorial exceptional prime. Its total transform has divisorial support equal to the entire inverse image of V\setminus U_V; every such divisor maps to a subset of codimension at least two by (5.25). Outside this support the resulting map \pi:\widehat V\to V is an isomorphism. Conversely a point on this support cannot be a local isomorphism point, because the divisor germ through it would map to a hypersurface contained in V\setminus U_V. Thus this is exactly the nonisomorphism locus. The complete support is globally simple SNC and includes every strict boundary component of V.

Each \pi-exceptional prime has center in V\setminus U_V. The preceding discrepancy argument shows that its log discrepancy is positive, proving property (2). Let \widehat Z be an irreducible intersection of k distinct transformed floor components. At its general point exactly those components occur, by SNC. If k=1, its prime valuation has log discrepancy zero. If k\geq2, the blowup of the smooth stratum has log discrepancy \operatorname{codim}(\widehat Z)-k=k-k=0. This is a prime on a projective resolution, so its image meets U_V. Since \pi is an isomorphism there, it is generically an isomorphism on that image. This proves property (3).□

Proof of Proposition 5.1. All statements except P\ne0 have been established. Suppose that P=0. Every component of P_Y, and hence every component of p^*M, has disappeared. On a common projective resolution with smooth source r:W\to Y, s:W\to V, set Q=r^*p^*M. It is effective and nonzero: the pullback of a nonzero effective Cartier multiple has a nonzero strict transform. It is s-exceptional. Indeed, a prime of W mapping onto a prime of V corresponds, by no extraction, to a surviving prime of Y; its coefficient in Q is zero under the supposition that every component of p^*M disappeared.

The actual rational line of Q is (pr)^*D. It is analytically nef on the compact Kähler W, by pullback of the metric inequalities for the original D. In particular it is relatively nef for the projective s. Exceptional negativity gives Q\leq0, contradicting its effectivity and nonvanishing. Thus P\ne0.□

Proof. Work in a relatively compact local embedding in \C^N, and refine the regular loci of Z, its singular locus, D, \Gamma, the boundary primes, and their intersections into a locally finite collection of smooth strata. After a neighborhood shrink only finitely many relevant strata meet the chosen compact closure. Choose a linear map \ell:\C^N\to\C^{n-2} whose differential is an isomorphism on the tangent space of \Gamma at a general smooth point where D is smooth. Such points exist because D is normal. For the restriction of \ell to each smooth stratum of dimension at least n-2, Sard’s theorem makes its critical values a measure-zero subset of the parameter space; strata of smaller dimension have measure-zero image [48]. The image of \Gamma contains a neighborhood of the chosen value. We may therefore choose a nearby value regular for all these restrictions and still meeting \Gamma at a general point. The corresponding level is transverse to every stratum it meets.

At regular points of Z the level is a smooth surface. The singular locus of a normal n-fold has dimension at most n-2; its strata therefore meet this level only discretely. Every irreducible component of the level has dimension at least two by the principal ideal theorem. It cannot be contained in that discrete singular intersection, and hence has dimension exactly two by its smooth dense part. The n-2 level equations have height n-2 at every point. In the Cohen–Macaulay local rings of Z they are a regular sequence. The quotient is consequently Cohen–Macaulay of dimension two. It is generically reduced and regular in codimension one, since its possible singular points are discrete; Serre’s criterion makes it normal. Transversality on D gives a smooth curve at the cut point, and the fact that d\ell is an isomorphism on T\Gamma makes this curve transverse to \Gamma.

Each boundary prime cuts an effective curve with its original rational coefficient, counted with its positive intersection multiplicity. At a generic codimension-one point of the normal surface, the ambient space and the slice are smooth and transverse to these primes. Dividing the ambient canonical form by the parameter volume identifies the restriction of the frame with the pluriadjoint frame of this sliced effective boundary. Both are rank-one reflexive sheaves on the normal surface, so the identification extends from codimension one and is an actual line identity.

To compare the orders, on the smooth D near a general point of \Gamma complete the parameters to coordinates (z_1,\ldots,z_{n-2},u) with \Gamma=(u=0). Write the meromorphic residue as

where h is a unit at a general point of \Gamma. Choose the regular level above also outside the proper zero set of its leading coefficient. Division by the base volume and restriction to that level then has order exactly c on the cut curve. On the ambient smooth locus the two operations commute: in coordinates with D=(x=0), both take the residue of h(x,u,z)(dx/x\wedge du\wedge dz)^{\otimes m} to h(0,u,z)(du)^{\otimes m} on the level, up to the ordering sign. The preceding reflexive identification makes the surface term the residue of the same actual frame. The restricted original residue is already a meromorphic form on the smooth curve C, so equality on its dense smooth-ambient part extends across the cut point. This proves the order assertion. In even degree the ordering sign is one.□

Proof. We induct on the number of indices. The assertions for the empty index set are the given normality, effective boundary, and crepant identity on V. Assume them for a stratum Z and all allowed weights there. Lower all unused floor weights. The resulting crepant SNC boundary on \widehat Z has every coefficient below one. The effective pair (Z,B_Z^{\mathbf e}) is therefore klt. Lemma 2.1 shows that Z has rational singularities and is Cohen–Macaulay.

To add an index j, retain only that unused weight at one and lower the others. The coefficient-one locus on \widehat Z is R=\widehat Z\cap\widehat S_j, a disjoint union of smooth components. Apply Lemma 2.1 to \pi_Z. Its connected fibers prevent the images of two such components from meeting. On each image D, the equality \pi_{Z,*}\OO_R=\OO_{\pi_Z(R),\red} identifies the proper bimeromorphic pushforward of the structure sheaf of its smooth resolving component \widehat D with \OO_D. Factoring through the finite normalization shows that D is normal. It is a compact analytic subspace of V, so it inherits a Kähler form by restricting local strictly plurisubharmonic potentials.

For arbitrary allowed weights retaining j, define B_D^{\mathbf e}=\pi_{D,*}\widehat B_{\widehat D}^{\mathbf e}. At this point it is a rational divisor whose effectivity remains to be shown. In a divisible even degree, a local frame of the ambient adjoint line pulls back and takes SNC residue to a meromorphic pluriform on \widehat D. The proper bimeromorphic map \widehat D\to D is an isomorphism at the generic points of divisors of the normal D. There the same map identifies the divisorial adjoint for B_D^{\mathbf e} with the restriction of the existing invertible line. Normal reflexive extension gives (6.3) on all of D. Pulling the line back to \widehat D recovers the original meromorphic map, since the maps agree on a dense open. This proves (6.2), including its meromorphic interpretation.

We verify that B_D^{\mathbf e} is effective. It is enough to test the coefficient at a general point of each prime in D, a codimension-two locus in Z. Apply Lemma 6.2, using the Cohen–Macaulay property already proved for Z. It reduces exactly that coefficient, with the actual residue order preserved, to the smooth cut curve in a normal surface germ with effective boundary. On a minimal resolution of this surface germ, write its crepant boundary as the effective strict transform plus an exceptional divisor E. For every exceptional curve C,

Indeed B^{\mathrm{str}}\cdot C\geq0, while surface adjunction gives K_{\widetilde T}\cdot C=2p_a(C)-2-C^2\geq0: exceptional self-intersections are negative and a smooth rational exceptional (-1)-curve is excluded by minimality. The negative-definite exceptional intersection matrix, with nonnegative off-diagonal entries, then implies E\geq0. Explicitly, if E=P-N has disjoint nonnegative parts and N\ne0, then E\cdot N=P\cdot N-N^2>0, contradicting E\cdot C\leq0 on each component of N. The strict transform of the smooth cut curve is finite birational over it and hence isomorphic. Adjunction to that smooth curve in a smooth surface with effective remaining boundary has a nonnegative coefficient. This is the coefficient originally tested. There is no coefficient to test when the new stratum is a point. Effectivity follows and completes this part of the induction.

For clarity, the full-weight pair on each stratum is dlt and has exactly the claimed floor. The SNC subboundary has coefficients at most one, so it is lc, and crepancy makes the pair on Z lc. On the smooth \widehat Z, let a divisorial valuation have center of codimension c, and choose local coordinates x_1,\ldots,x_c,\ldots at a general point of that center, with the SNC boundary components among the coordinate divisors. Give an unmarked coordinate weight zero and write the boundary weights as b_i\leq1. The reduced coordinate arrangement is lc, so

All v(x_i) here are positive. A zero discrepancy therefore forces every b_i=1; its center is an intersection of coefficient-one components. Each such intersection has image meeting the isomorphism locus of the original special resolution. In particular each coefficient-one divisor is nonexceptional for \pi_Z; the others have coefficients below one. Let F_Z\subset Z be the complement of the largest open over which \pi_Z is an isomorphism. It has codimension at least two on the normal target Z, and contains no entire image of a zero-discrepancy center. Its inverse image on \widehat Z may already be divisorial. If necessary, principalize the pulled reduced ideal \mathscr I(F_Z)\OO_{\widehat Z}, and then apply the ordered-boundary resolution as in Lemma 5.6, marking each distinct prime in the prior SNC support, the newly principalized support, and every divisorial exceptional prime. All new exceptional centers lie over F_Z. The displayed inequality makes their discrepancies positive. This is a dlt resolution in the standard sense. It also proves that the coefficient-one primes on Z are exactly the images of the one-index intersections.

It remains to check the asserted descent of sections. With full weights, let R=\widehat B_{\widehat Z}^{=1}. The preceding floor identification gives (\pi_Z(R))_{\red}=C_Z; write \pi_R:R\to C_Z for the induced restriction of \pi_Z. Sections on the incident strata pull back to sections on its smooth components. Residue compatibility means equality on the entire scheme-theoretic pairwise intersections, which are reduced smooth SNC strata. The local equalizer for an SNC union therefore glues them to a section on R; it is the elementary equalizer for the coordinate ideals of its components. Even-degree iterated residues make all paths to deeper intersections identical. By Lemma 2.1 and projection formula,

for any ambient Cartier line L under consideration. The section descends uniquely to all of C_Z. This reasoning applies also to Z=V and C_Z=S. In particular it retains the conductor coefficient at a general double crossing on each normalized component; it does not posit an additional conductor boundary on the nonnormal union itself.□

Proof. The equality with the ideal holds because on the normal T both sheaves consist of holomorphic functions vanishing at every height-one prime in C. Work on a small connected base open trivializing the line whose pullback is m(K_T+G). Its nonvanishing pulled-back frame is a local meromorphic m-pluricanonical form \theta.

Adjoin the full set of mth roots of this form. This construction is local even without a canonical Cartier frame. At a normal local domain A, choose a meromorphic canonical generator \eta and write \theta/\eta^m=a/b. The monic algebra

is finite and reduced; its analytic normalization is finite [34, Part B, Section 4, Corollaries 2–3]. Changes of meromorphic canonical generator multiply the root coefficient by an mth power, and identify the total algebras and their integral closures. They therefore glue to the normalized cover. Keep every component and the full \mu_m-action. The tautological form \tau=t\eta is a meromorphic top form on a projective resolution with smooth source; take this resolution functorially and equivariantly, after shrinking near the compact fibers. Write l:H\to T for the composite.

At a general prime of T with boundary coefficient b, the order of \tau upstairs is

where e is the ramification index. This follows from the pole order -eb of the root coefficient and the differential ramification order e-1. It is -1 when b=1 and is an integer between zero and e-1 when 0\leq b<1. A meromorphic form in the \tau-character has the form h\tau, where h is an invariant meromorphic function and hence descends to T; this uses the entire root algebra, whose invariants are the normal base. Regularity in codimension one forces h to be holomorphic and to vanish on C.

Conversely, suppose h\in\OO_T(-C). On a log resolution of the pair, every crepant coefficient is at most one. A coefficient-one exceptional prime has center in C, because the pair is klt away from C; the pullback of h has positive order there. Thus in SNC coordinates the density |h|^2|\theta|^{2/m} has every exponent strictly above the integrability threshold. It is locally integrable. Change of variables makes l^*h\tau locally square integrable on the smooth resolution, and a square-integrable meromorphic top form is holomorphic. This proves the exact character equality

Each component of the full normalized cover dominates the normal base, and each resolved component does so as well. The inverse image in T of the connected base open is connected, because f is proper with connected fibers; normality makes it irreducible. Finite maps followed by projective resolutions have Kähler sources near the whole inverse image of a sufficiently small base neighborhood: patch relative ample metrics and add a large pulled-back Kähler form. Canonical torsion-freeness [21, Theorem 2.9 and Proposition 2.11] applies componentwise. For the generically finite map to T, its higher canonical images vanish generically and hence vanish. For the map to P, its first canonical image is torsion-free. Leray and the character projector in (7.3) make R^1f_*\OO_T(-C) a direct summand of that torsion-free sheaf. This proves the assertion locally, and hence on P.□

Proof.

Choose a small rational \varepsilon>0 for which B_0-\varepsilon C_0 is effective klt. This uses dlt and the fact that all its lc centers are contained in the floor. Let H be an integral very ample divisor on Y and choose a rational effective H'\sim_\Q dH with d>2\dim Z such that

is still klt. A sum of sufficiently many general members of |H| with small rational weights does this by Bertini on a log resolution. If Y is a point take H'=0.

The adjoint of this perturbed pair is not pseudo-effective. On a smooth general fiber of a resolution of f_0, its class is the negative of the nonzero effective divisor \varepsilon C_0 restricted to that fiber: the full adjoint and H' are pulled back from Y, and the floor is horizontal. Its integral against a Kähler power is strictly negative. If the ambient class were pseudo-effective, the potential of a positive current representing its pullback would restrict to a positive current on almost every such smooth fiber, contradicting this integral. The smooth horizontal-fiber conditions hold on a nonempty open, so the almost-everywhere restriction is sufficient.

This same integral also produces a negative class in the cone used by the Kähler cone theorem. If the smooth fiber has dimension v>0, push its positive closed bidimension-(1,1) current \omega_F^{v-1} through the fiber inclusion and the resolution to Z_0. It represents a class in \overline{\operatorname{NA}}(Z_0), and its pairing with the perturbed adjoint is exactly that strictly negative integral. The cone decomposition therefore supplies a negative extremal ray.

Run the klt Kähler program for this perturbed adjoint. In dimension three, the negative extremal-ray and contraction theorems [12, Corollary 5.3 and Theorem 5.5] are used through Subsection 4.4, which incorporates Proposition 4.3 and the specified external results. They give connected projective contractions to normal compact Kähler targets. The supporting class differs from the klt adjoint by a Kähler class, which is the projectivity condition in that theorem. The supporting class need not be big, and the perturbed adjoint here is not pseudo-effective. Flips and termination of flip sequences are provided by [11, Theorems 2.24–2.25]. In dimensions at most two the non-pseudo-effective pair is projective and the ordinary projective program applies. For a surface, otherwise a non-Moishezon Kähler resolution has a nonzero holomorphic two-form: if H^{2,0}=0, rational approximation of a Kähler class would make it projective. That form descends reflexively and, with the effective boundary, gives a section of a positive adjoint multiple, contradicting non-pseudo-effectivity. A Moishezon Kähler klt surface is projective by Namikawa’s criterion [40, Corollary 6’]. Curves are projective.

We specify the ordinary \Q-factorial and finiteness details used by this program. For a projective negative extremal contraction, a global rational line of degree zero on all contracted curves descends in a multiple by [12, Theorem 2.44(3c)]. Here we apply its projective relative theorem to the effective klt pair, with the compact set equal to the entire compact target. For this projective surjection, property Q of [12, Definition 2.42] requires a normal source and compact source and target, as here. All fiber curves span the one relative ray, so the theorem’s contraction is the given contraction. Indeed its map is constant on each connected projective fiber of the given contraction, and rigidity factors it through that contraction. Its structure map back to the relative base supplies the reverse factorization; surjectivity makes the two factorizations inverse. This use of Theorem 2.44 does not assert a factoriality hypothesis for that theorem. At a divisorial step, adjust the transform of a destination Weil divisor by a \Q-Cartier exceptional prime of nonzero degree on the ray, and descend the resulting degree-zero line. Comparison in codimension one makes the destination divisor \Q-Cartier. At a flip, adjust on the negative side by the negative adjoint, descend, and pull to the positive side; the positive adjoint is \Q-Cartier and the same comparison applies. Boundary components and the adjoint then also give the canonical reflexive power. This preserves ordinary \Q-factoriality.

The rank of global rational divisor classes modulo curve numerical equivalence is finite. Each running model is a connected compact complex analytic space. Its reduced underlying real analytic space has bounded Zariski tangent dimension by a finite analytic chart cover. Acquistapace–Broglia–Tognoli [1, Theorem 1] embed it closedly in Euclidean space, and Łojasiewicz [41, Theorem 1] gives a compatible locally finite triangulation; the subcomplex corresponding to the compact model is finite. Hence its rational H^2 is finite-dimensional. The first Chern class sends global rational Cartier divisors to that group. Every divisor in its kernel has degree zero on each compact curve, by restriction to the curve’s normalization. The rational divisor space modulo curve numerical equivalence is consequently a quotient of the Chern-class image, and has finite rank. The rank drops at a divisorial contraction, by the nonzero exceptional ray degree, and does not increase at a flip. For the latter assertion a numerically trivial line on the negative side descends as above; the descended line is curve-numerically trivial because every curve downstairs is covered by a curve upstairs, using projectivity over that curve. Its pullback is trivial on curves on the positive side, and every divisor there is a transform by smallness. There can therefore be only finitely many divisorial steps. Together with termination of flip sequences this gives a finite program to a nef model or a Mori contraction. All steps extract no divisors.

Every step factors over Y. Inductively write the running perturbed adjoint as

For a negative contraction apply the relative cone and length theorem [12, Theorem 2.45] to K_{X_i}+B_i-\varepsilon C_i over its entire compact contraction target. The source is ordinary \Q-factorial and the pair is klt. Property Q holds as above; because the compact set is the whole target, Theorem 2.45’s factoriality over that set is precisely the global factoriality of the source, including its canonical reflexive power. A negative generator \Gamma has

If f_i(\Gamma) were a curve, then f_i^*H'\cdot\Gamma\geq d, contradicting the negativity of A_i^{\varepsilon}. On the nonnegative part of that relative cone, both K_{X_i}+B_i-\varepsilon C_i and the nef f_i^*H' are nonnegative, while their sum is nonpositive on the contracted cone; its H'-degree is zero there as well. All contracted curves are vertical over Y. A holomorphic map from a connected projective fiber that is nonconstant would map a curve nontrivially, so f_i is constant on each fiber. Rigidity and normality of the target give the factorization, and the positive side of a flip factors through the same base.

We check the boundary properties needed to repeat the step. No prime is extracted, so the transformed full boundary is effective and its floor is exactly the transform C_{i+1} of C_i. The running MMP preserves the klt pair B_i-\varepsilon C_i+f_i^*H'. Removing the effective divisor f_{i+1}^*H' shows that B_{i+1}-\varepsilon C_{i+1} is klt. In particular the full pair is klt away from its floor.

The full adjoints remain actually crepant pullbacks from Y. Indeed the chosen meromorphic pluriadjoint identity extends to the new model in codimension one, since no prime is extracted, and then reflexively. On a common graph both meromorphic maps start from the same line pulled back from Y and agree on the dense isomorphism open. They agree everywhere as meromorphic maps. Equality of discrepancies therefore preserves log canonicity of the full pair. If r,s are the graph projections, negativity for the perturbed step gives

A horizontal component of C_i cannot disappear into an entirely vertical C_{i+1}, since the coefficient of the right side along its strict valuation would then be negative. Thus the floor remains horizontal. On every subsequent model the full adjoint and H' still come from Y. Repeating the resolved smooth-fiber integral and pushing its Kähler-power current as above gives a negative class in its \overline{\operatorname{NA}} cone and hence a negative extremal ray. In particular a nef endpoint is impossible.

We have obtained a Mori contraction

with T normal compact Kähler, u projective of fiber type, and both maps having connected fibers. For g, this follows also from (g u)_*\OO_T=\OO_Y and u_*\OO_T=\OO_W. The full boundary G is effective lc, its adjoint is an actual pullback from Y, and it is klt away from C^+=\floor G: subtracting \varepsilon C^+ is klt. The running perturbed adjoint satisfies

The left side of the second equivalence is relatively ample for the Mori contraction. Thus C^+ is relatively ample for u. The graphs of the divisorial steps and flips are projective over both sides. A main component of their iterated fiber product, followed by normalization and projective resolution, gives the common projective resolution in the statement. Its smooth source is compact Kähler.□

Proof. If C_Z is vertical over Y_Z, the consequence of (7.4) gives extension in a uniform large-degree tail without a comparison. Suppose therefore that it is horizontal, and use the small model q_Z and the Mori model u:T\to W, g:W\to Y of Lemma 7.2. Extending the pulled-back section on C_0 suffices: a section on Z_0 descends to Z, and equality of its restriction after pullback implies equality on the reduced C_Z, since every component is covered birationally.

The floor on general Mori fibers

We now compare the connected components of a general floor fiber with the markings of one Mori fiber. On a common projective log resolution of the program, the full crepant boundary is the same on both sides. Its coefficient-one union maps to both C_0 and C^+ with connected fibers by Lemma 2.1. After restriction over any y\in Y, proper closed maps with connected fibers preserve connected components. Thus the connected components of the underlying spaces of C_{0,y} and C^+_y correspond. This is a topological statement, not a reducedness claim for their scheme fibers.

The finite Stein space of C^+\to W has every component dominating W. Indeed relative ampleness makes C^+\to W surjective, and Lemma 7.1 and (7.4) give the torsion-free property used in the preceding discussion. For general y, every irreducible component of the finite Stein fiber has dimension \dim W-\dim Y, by the dimension theorem. The fiber W_y is irreducible of that dimension: a resolution of W has connected fibers over Y, and a general one is smooth and connected, hence irreducible, and surjects onto W_y. Each component of the finite Stein fiber therefore dominates W_y. It follows that every connected component of C^+_y meets u^{-1}(w) for a common general w\in W_y.

The underlying space of a general fiber F of u is irreducible by the same resolution argument. If \dim F\geq2, a Cartier multiple of C^+ restricts to a nonzero effective ample Cartier divisor on F_{\red}. Its support is connected. To recall the reason, if an ample effective divisor split into two disjoint nonzero parts, general hyperplane sections would reduce to an irreducible projective surface F_2. Write the two induced nonzero effective Cartier parts on it as D_1,D_2, and let v:\widetilde F\to F_2 be a projective resolution with smooth source. Their pullbacks are effective and orthogonal. Although v^*(D_1+D_2) is only nef and big, the projection formula gives

the pushforward is a nonzero effective curve cycle and D_1+D_2 is ample on the integral surface. Orthogonality then gives (v^*D_i)^2>0 for both i. The surface Hodge index theorem forbids two such orthogonal positive classes. This argument does not require normality of F_{\red}.

If \dim F=1, choose F also in the generic-smoothness open. Generality avoids the singular locus of T, the singularities and intersections of the boundary, and the ramification of its horizontal primes. The fiber is a smooth connected curve, and the full pullback identity gives

There is at least one coefficient-one point. Hence F\simeq\PP^1, and its floor has one or two points. In the two-point case these points exhaust the horizontal boundary on F. In all connected cases, every connected component of C^+_y meets the same connected subset C^+\cap F, so C^+_y, and therefore C_{0,y}, is connected. Then no comparison is needed. In the two-point case each connected component of C_{0,y} is represented by at least one of these two markings, after their transfer through the common graph.

The two-marking comparison

Normalize each surviving horizontal prime D\subset C^+ and take the Stein factorization

The first map is projective bimeromorphic and E_D is normal; the second is finite. If there are two horizontal primes, each finite map has degree one and is an isomorphism over the normal W. The main component of the fiber product of their normalizations over W gives a proper bimeromorphic comparison. If there is one horizontal prime, E_D\to W has degree two. On a dense open it is étale and has an exchange. The reduced horizontal non-diagonal component of E_D\times_W E_D has finite generically one-to-one projections to the normal E_D; both are isomorphisms. It defines the exchange involution globally, including across the branch. The main component of D^\nu\times_{E_D,\mathrm{exchange}}D^\nu lifts it to a proper bimeromorphic graph. It is the Stein space of the normalized prime that is used here, not the possibly nonnormal Stein space of the entire floor.

Compose this graph with the common graphs of the program and with the small model. It gives the asserted comparison between normal primes of C_Z, possibly a self-comparison. Each surviving prime is generically finite over W; the image in W of the proper subset where its transfer is not an isomorphism is proper. Thus both markings on a general Mori fiber lie in the common transfer isomorphism loci.

These graph projections are projective. The maps D^\nu\to E_D are projective and the maps E_D\to W are finite; their fiber products and closed main components are therefore projective over the branches. The small-model and MMP graphs have the same property. Iterated main components and finite normalization preserve it, so the comparison admits a projective resolution with smooth compact Kähler source.

We verify the actual meromorphic identity required by property (1). The two residues of a logarithmic fiber form give the same comparison as the even Poincaré-residue diagrams of [37, Section 3, Definition 13 and Proposition 14]; we include the calculation for the normalized two-branch construction above. On a smooth ruled open of W, order the two markings after an étale local cover when necessary, choose a base volume form \xi, and a fiber coordinate with markings at zero and infinity. The full q-pluriadjoint frame coming from Y has the form

There is no other horizontal divisor on the general fiber. Its two residues are a(w)\xi^{\otimes q} and (-1)^q a(w)\xi^{\otimes q}; they agree because q is even. The calculation is invariant under exchanging the two local markings. Full crepancy transfers it to the original primes. On a resolution of their comparison graph, both actual adjoint lines are the same pullback of N_Z from Y. Their meromorphic embeddings agree on this dense ruled open, hence agree everywhere. Thus the invertible subsheaves of meromorphic pluriforms are equal even at exceptional primes. An abstract Q-linear equivalence without this residue calculation would not suffice.

A section satisfying this comparison has equal values at the two representatives over a general w. On every connected component of a compact reduced floor fiber it is constant in the trivialized pulled-back line: holomorphic functions on a compact connected reduced complex space are constant. The representatives meet every component, so its values agree on C_{0,y} for every y in a dense base open. The horizontal consequence of (7.4) extends it to Z_0, and then it descends to Z. The only asymptotic bound on k came from Serre vanishing in the vertical case. Finitely many strata admit a common sufficiently divisible tail. This proves all three properties.□

Proof. Let i:Z\hookrightarrow V be the parent stratum, and choose a common projective resolution with smooth compact Kähler source U, with maps a:U\to Z and b:U\to T to its original and Mori models. Use the single class

Resolve the marked-branch comparison graph and its maps to U and the minimal surfaces. We obtain a smooth compact Kähler surface D with maps r_i:D\to U, bimeromorphic maps d_i:D\to P_i, and bimeromorphic maps to the original Z_i, such that a r_i is the inclusion of that original branch after its resolution and d_2=\tau d_1. These identities hold on the common marked open and hence everywhere. This construction includes a self-link: its two maps to the same original prime may differ by the normalized exchange. Resolution independence and projection formula give

The exceptional image of the modification b:U\to T has codimension at least two in the normal threefold, and hence dimension at most one. The Mori base in this two-marking case has dimension two. After removing its image, the discriminant, and the singular base locus, U is therefore a smooth proper \PP^1-fibration over a dense open of the base. For such a fibration \pi:U^\circ\to W^\circ, the differential sequence and H^0(\PP^1,\Omega^1)=0 give

Indeed the quotient of \Omega^2_{U^\circ} by \pi^*\Omega^2_{W^\circ} is \pi^*\Omega^1_{W^\circ}\otimes\Omega^1_{U^\circ/W^\circ}, whose direct image is zero, while \pi_*\OO_{U^\circ}=\OO_{W^\circ}. Thus the two restrictions of a holomorphic two-form on U agree on the dense paired-branch open of D, even for the double branch after an étale local ordering. They agree on all of D.

Holomorphic pullback and proper pushforward on compact Kähler manifolds preserve Hodge type. The map

is therefore a rational Hodge morphism; pushforward here is between equal-dimensional surfaces. The preceding equality kills its (2,0) part and, by conjugation, its (0,2) part. Its rational image is of type (1,1), hence belongs to \NS(P_1)_\Q by Lefschetz (1,1). Extend scalars to \R in (8.4) to obtain (8.3).□

Proof. The intersection form on H^{1,1}(P,\R) has Lorentz signature (1,h^{1,1}-1). Its restriction to N is nonpositive. To see this, a rational positive-square class in N, after scaling and choosing its sign, is c_1(L) with positive Kähler degree. Riemann–Roch and Serre duality give h^0(P,L^v)\gg v^2: the Euler characteristic has positive quadratic leading term and H^0(P,K_P\otimes L^{-v})=0 for large v by its negative Kähler degree. This would make P Moishezon and hence projective. Rational approximation in N rules out a positive real class as well.

If N is negative definite, orthogonally project c to N^\perp. The projection c_\perp has positive square and is fixed by every \sigma^*: the congruence kills its projected difference, and each automorphism preserves N and the intersection form. The perpendicular of c_\perp is negative definite, so all eigenvalues on H^{1,1}(P,\R) have modulus one.

If N is degenerate, its radical is a rational isotropic line \ell; nonpositivity in a Lorentz space permits no larger radical. An integral automorphism preserves a primitive integral generator of \ell up to sign. The invariant flag

has eigenvalues \pm1 on the first and last quotients, which pair dually, and a negative-definite middle quotient \ell^\perp/\ell. All (1,1) eigenvalues again have modulus one. This argument permits unipotent action and makes no finiteness assertion for all cohomology.

Write \sigma^*\eta=\delta_\sigma\eta. Integration of \eta\wedge\bar\eta gives |\delta_\sigma|=1, and the conjugate eigenvalue has the same modulus. Since a nowhere-vanishing canonical form spans H^{2,0}(P), all eigenvalues on H^2(P,\C) now have modulus one. The action on H^2(P,\Z)/\mathrm{torsion} is integral. Kronecker’s theorem makes each eigenvalue a root of unity; if its order is n, its cyclotomic polynomial has degree \varphi(n)\leq b_2(P). There are only finitely many such n. The scalars of every element of G therefore lie in one finite set of roots of unity, which proves finite image on \C\eta^{\otimes m}.□

Proof. For curves, the comparison is an isomorphism preserving the marked points and their residue generators. Invariance makes the lower point values equal, so the assertion follows.

For surfaces, take a target floor curve and view its valuation on the source of the comparison. If its center is a floor curve there, adjunction of the equality of meromorphic adjoint subsheaves on the graph gives a crepant residue comparison of the two curve pairs. This is an arrow of \mathcal G_1, and the assigned curve tuple is invariant.

If the center is a point, it is a zero-dimensional lc center of the source dlt surface, by crepancy. It is one of the point strata and is an actual smooth crossing of two coefficient-one curves: the resolution used in Proposition 6.3 is an isomorphism at such a point. In coordinates (x,z) for the crossing, a divisorial lc place above it arises by successive blowups of crossings of two coefficient-one branches. To verify this, factor a surface resolution into point blowups. In an SNC surface boundary the new crepant coefficient at the blowup of a crossing is the sum of the two coefficients minus one; at a point on just one branch it is that coefficient minus one. Starting with coefficients at most one, a new coefficient one can arise only at a crossing of two coefficient-one branches. This remains true at every subsequent step.

The corresponding exceptional curve is a \PP^1 whose different has exactly the two adjacent coefficient-one points. Residue of (dx/x\wedge dz/z)^{\otimes qk} along it is (dt/t)^{\otimes qk}, up to a sign removed in the even degree: at each crossing blowup the logarithmic change-of-coordinate determinant is \pm1. Further point blowups do not change this meromorphic residue on its strict transform. The normal target floor curve is bimeromorphic, hence isomorphic, to this \PP^1, and its actual log line is trivial with that generator. Its section is determined by the common residue at the two point strata. Pulling a local surface section through the crossing restricts on the exceptional curve to its value at the point times that generator. Compatibility of the original tuple identifies this value with its assigned point component, and invariance makes it the assigned value at both target point strata. Thus the entire target curve receives exactly the prescribed lower section.

The two alternatives apply to every composite surface comparison, so the assertion holds for the whole groupoid, not only for a generating link.□

Proof. For point strata take the same scalar in the canonical zero-form generator 1 on every point. The even iterated residue convention identifies these generators along all paths. This gives an invariant generating system in dimension zero.

Suppose a system has been constructed through dimension d-1, for 1\leq d\leq3. Replace it by powers and their span in a sufficiently large common degree, still denoted k. For a d-stratum Z, the components of any lower tuple glue to a section on its entire floor C_Z by Proposition 6.3. They satisfy its link comparison: for d=1 the link is a point comparison, for d=2 a curve comparison, and for d=3 one of the generating surface links. Lower invariance supplies each equality. Proposition 7.3 therefore extends every such boundary section to Z, once the degree is sufficiently large. There are only finitely many strata, so the degree can be chosen uniformly.

Let \mathcal P_d(k) be the vector space of all tuples through dimension d whose lower part belongs to the chosen lower system and whose d-components restrict to those glued floor sections. It maps surjectively to the lower system, since the finitely many extensions can be chosen independently. This is a linear space of lifts; no choice of a nonlinear extension operator is involved.

The pre-system \mathcal P_d(k) generates. For z\in Z with y=f_Z(z)\in f_Z(C_Z), choose a floor point above y and a lower tuple nonzero there. Any extension to Z is the pullback of a section of N_Z^k, by (7.1). Its value at y is nonzero, and hence it is nonzero at z. If y\notin f_Z(C_Z), the sheaf \mathscr I_{f_Z(C_Z)}\otimes N_Z^k is generated for large k by Serre’s theorem. A section nonzero at y pulls back to a section vanishing on C_Z; with zero lower tuple and all other new components zero, it belongs to \mathcal P_d(k). This also covers an empty floor. Lower points remain generated because the projection to the lower system is surjective.

For d=3 this pre-system is the required final system. For d=1,2 we impose invariance by norm products. Lemma 9.1 applies to every pre-tuple: all transported top components have the same assigned lower restrictions. A product of h such factors will therefore have the original lower tuple raised componentwise to h.

For each orbit of d-strata choose a representative Z, and let G_Z be the finite isotropy image on H^0(Z,L_Z^k) from Proposition 8.1. Given a pre-tuple with component s_Z, form its norm

Choose a common integer h divisible by every |G_Z|, and use P_Z(s_Z)^{h/|G_Z|}. Transport this section to every member of its orbit. This is well defined: a change of the chosen transport by an isotropy arrow permutes the factors in degree k, and transport commutes with multiplication. No finiteness statement about the possibly larger representation in degree kh is needed. Each factor has the assigned lower restriction proved above, so the resulting lower tuple is the original lower tuple raised componentwise to h. Thus all these sections are compatible across different orbits and all deeper intersections, and they are invariant in dimension d.

They still generate. Fix a point z of an orbit member. For each of the finitely many transported factors, choose a point over z on a resolution of its comparison graph. Equality of the pulled-back invertible adjoint subsheaves identifies the line fibers. Nonvanishing of that factor at z is therefore the nonvanishing of the representative pre-section at a definite point of Z. Each such condition is a nonzero linear evaluation functional on the generating vector space \mathcal P_d(k). A finite union of the proper kernels of these functionals cannot cover a complex vector space. One pre-tuple satisfies all of them, and its norm is nonzero at z. At a lower point choose a pre-tuple whose assigned lower value is nonzero; its hth power remains nonzero. Finally take the linear span of all the constructed norm tuples. Compatibility and invariance are linear conditions, so this span is an invariant generating system in degree kh.

This completes the induction through dimensions one and two; the pre-system at dimension three then has all the required properties.□

Proof of Theorem 6.1. Apply Proposition 9.2. Its three-dimensional components agree on every subordinate stratum, so Proposition 6.3 descends each tuple uniquely to a section of L^k|_S. For any x\in S, choose a component through x. The generating system has a tuple whose value there is nonzero. It is the pullback of the descended value in the actual line fiber at x, so that descended value is nonzero. Nakayama’s lemma makes the evaluation map onto L^k|_S surjective at x. The finite-dimensional span of these global sections therefore generates at every point, proving semiampleness on the whole reduced floor.□

Proof. The analytic Kummer sequence

is exact also for singular or nonreduced analytic spaces. A unit germ has a local holomorphic root; the kernel is the locally constant sheaf of rth roots of unity, since u^r-1 has distinct roots, including in a local ring with nilpotents. Line bundles are classified by H^1(\OO^\times), and abelian torsors by H^1 of the corresponding sheaf [54, Tags 09NU and 02FQ].

For an abelian sheaf F and a compact subset whose distinct points have disjoint neighborhoods, SGA 4 continuity [51, Exposé Vbis, Lemma 4.1.3] gives

The separation hypothesis holds in the Hausdorff analytic space. Although the intrinsic structure sheaf of A need not be the inverse image of \OO_X, the inverse image of \mu_r is exactly \mu_r on A. Naturality of Kummer therefore makes the obstruction to a root of L restrict to zero in H^2(A,\mu_r), because Q is such a root there. Continuity in degree two kills that obstruction on a smaller neighborhood, giving a root pair (P,\alpha).

The difference between (P|_A,\alpha|_A) and (Q,\beta) is a \mu_r-torsor. By continuity in degree one it extends to a torsor on a smaller neighborhood. The associated line with its trivialized rth power twists P to give the prescribed pair on A. Finally, the sheaf of compatible isomorphisms of two root pairs is itself a locally trivial finite \mu_r-torsor. A prescribed section on A trivializes its restriction there. Continuity in degree one therefore trivializes this torsor on a smaller neighborhood; choose a section there. The ratio of its boundary restriction to the prescribed section is a section of \mu_r on A. Continuity in degree zero extends that ratio uniquely as a germ, and correcting the chosen section gives the prescribed extension. The same degree-zero injectivity gives uniqueness as a germ. These sections are holomorphic, being locally solutions of z^r=u for a holomorphic unit.□

Proof. Local finiteness and compactness give an open neighborhood \mathcal N_1 of the central fiber which meets only finitely many labels. Remove from the base the closed image of H\setminus\mathcal N_1; properness then puts the entire restricted support in \mathcal N_1. There are now finitely many closed indexed intersections to consider. The connected components of each form a locally finite family in H: the intersection is closed and smooth, so near one of its points a small smooth neighborhood meets only its own component, and near a point outside it a neighborhood avoids it. Compactness, followed by the same properness argument, gives a second neighborhood \mathcal N_2 meeting only finitely many such components and a base restriction on which the whole support lies in \mathcal N_2.

Let C_1,\ldots,C_N be the whole connected components of the pre-restriction closed intersections which meet \mathcal N_2. We retain the whole components, not just their intersections with \mathcal N_2. Each C_a is closed in a closed stratum and is a connected complex manifold. Thus f_a=h|_{C_a} is proper over the preceding base and C_a is irreducible. Their restrictions cover every closed intersection after the current restriction, including the singleton intersections.

Put S_a=f_a(C_a) with its reduced structure. Remmert’s proper mapping theorem makes S_a closed analytic [25, Section 7, no. 1, Satz 1, p. 60], and it is irreducible because C_a is. Inside any prescribed open coordinate neighborhood V\ni t_0 contained in the preceding restrictions, take the locally finite irreducible decompositions of all S_a\cap V. Let B be the union of those components which do not contain t_0. A union of a subfamily of the locally finite irreducible components of an analytic set is closed analytic. Since the list of a’s is finite, B is closed analytic in V and misses t_0. Set U=V\setminus B.

This removal does not split any retained irreducible component A of an S_a\cap V. Its regular locus is connected, and its intersection with B is a proper analytic subset: otherwise closedness and density would imply A\subset B, contrary to t_0\in A\setminus B. The complement of a proper analytic subset in a connected complex manifold is connected. One may see this by perturbing a path, in a finite chain of coordinate balls, off a subset of positive complex codimension. Hence \operatorname{Reg}(A)\setminus B is connected and A\cap U is irreducible. Only finitely many components of S_a\cap V contain t_0, by local finiteness. Their restrictions give a finite cover of S_a\cap U by irreducible closed analytic subsets, all containing t_0. Therefore every irreducible component of S_a\cap U contains t_0; if S_a misses t_0, its restriction is empty.

Let W be a connected component of f_a^{-1}(U). It is open in the connected smooth C_a, is closed in f_a^{-1}(U), and is smooth and irreducible. Its map to U is proper. Put r_a=\dim S_a. The maximal-rank locus of f_a is dense in C_a: the generic rank theorem gives maximal rank r_a, and the lower-rank locus is a proper analytic subset defined by the maximal minors. The nonempty open W meets the maximal-rank locus. Its proper image is consequently an irreducible closed analytic subset of S_a\cap U of dimension r_a. Every component of S_a\cap U has this dimension because S_a is irreducible and hence pure-dimensional. The image is therefore an irreducible component, since a proper analytic subset of such a component has smaller dimension. It therefore contains t_0, and W meets f_a^{-1}(t_0).

The compact analytic fiber has finitely many connected components: on its reduction, irreducible components are locally finite and compactness makes their number finite. Each W\cap f_a^{-1}(t_0) is a nonempty open-and-closed subset of this fiber. Different W’s give disjoint such subsets, so

This includes zero-dimensional and other vertical images and permits nonreduced scheme fibers.

Reindex the singleton strata by these finitely many components W. An intersection of the new labels is an open-and-closed part of the restriction of the corresponding old closed intersection, hence is a union of some of its finitely many connected components. It remains smooth and proper, and the relative form restricts. Distinct new pieces from one old label are disjoint, so the local SNC branches and their ordering are unchanged. This proves the assertion.□

Proof. Let v be a local meromorphic frame of \OO_Z(aE). By (10.5) and aq=r, the q-pluriform v^q\pi^*s is, up to a holomorphic unit, the pullback of a local frame \ell_V of L. On a log resolution of (V,B,G), write b_F for a crepant boundary coefficient and m_F=\ord_F(s/\ell_V). Then m_F\geq0, b_F\leq1, and b_F=1 implies m_F>0: the dlt pair is klt off S, and the section vanishes exactly on S. Thus the local density

has exponent -b_F+\epsilon m_F>-1 at each prime for every \epsilon>0, and is locally integrable. Change of variables under the proper generically finite map \pi\rho preserves this integrability. For \alpha=(\rho^*v)\tau, let k_F be its integral order at a prime upstairs and n_F the order of the pulled section ratio. Integrability says k_F+\epsilon n_F>-1 for all \epsilon>0. The order n_F is positive exactly on H. Hence k_F\geq-1 on H and k_F\geq0 elsewhere. This proves the logarithmic map. Cartier adjunction on the reduced SNC divisor gives its residue map on H, with \rho_H^*\mathcal A_{-a}=\rho^*\OO_Z(aE)|_H.

Let \chi be the \tau-character of the second cover. At a general point of E over S_i, differential pullback gives

The second root is unramified there and \ord_E\tau=a-1. At any other codimension-one prime the first cover is étale and the original boundary coefficient is \beta\in[0,1). For e=q/\gcd(q,q\beta), the second-cover order is e(1-\beta)-1\in\{0,\ldots,e-1\}, by (7.2). A meromorphic form in character \chi is f\tau for an invariant meromorphic f descended to Z. The orders force \ord_E f\geq-(a-1) and nonnegative orders at all other primes. Normality extends it as a section of the indicated invertible line. Conversely, the logarithmic map multiplied by the equation of E has no pole because D_E\geq H. We obtain the exact sheaf identities

The second is projection formula for the invertible line \OO_Z(E).

Canonical torsion-freeness [21, Theorem 2.9 and Proposition 2.11] applies locally near E to each dominating resolved component. More precisely, around a compact fiber choose the Kähler neighborhood just constructed. Properness of \rho supplies a smaller normal target neighborhood whose entire inverse image lies in it, by removing the closed image of its complement. After a base shrink this neighborhood contains all of E in the root chart. The restricted morphism is proper with Kähler smooth source, so the cited theorem applies there. These neighborhoods cover E. On each, the higher direct images for the generically finite resolution vanish because they vanish generically. Finite pushforward is exact. Thus on their union Z^\circ\subset Z, an open neighborhood of E,

Projection formula gives the same restricted vanishing for \omega_{\widehat Z}(D_E).

Put A_E=\mathcal A_{-a}=\OO_E(aE). Cartier adjunction for the possibly nonreduced divisor D_E is the exact sequence

Since D_E is the scheme inverse image of E, it has a morphism \rho_D:D_E\to E. The preceding vanishings and character identities identify

One can see this first after exact closed pushforward from E to Z^\circ, where it is the quotient of (10.14) by (10.13). Closed pushforward detects its cohomology sheaves, and the canonical truncation of a complex concentrated in degree zero gives (10.15) on E. No derived full-faithfulness assertion is needed.

Because A_E is invertible, its pullback followed by the residue map defines A_E[0]\to R(\rho_H)_*\omega_H. The inclusion \omega_H\hookrightarrow\omega_{D_E}, followed by the character projection and (10.15), gives a map back to A_E\tau[0]. On degree zero their composition sends f to f\tau in the quotient of the two character lines. Under the chosen identification it is the identity. An endomorphism of the sheaf A_E[0] in degree zero is determined by its sheaf map, so this is a derived retraction. Applying Rg_* proves the split injections (10.12). All maps were defined by the tautological form, the actual divisor ideal, and residue, so they respect root comparisons. Under an ordinary product the tautological form is relative to the new factor; wedging with a local frame of its canonical line gives the absolute map and the stated \omega_{B_0} factor.□

Proof. Choose finitely many pairs of parameter opens U_i'\Subset U_i, with the U_i' covering f(S), such that a local root chart and its proper map h_i:H_i\to U_i are defined on U_i. Each h_i^{-1}(\overline{U_i'}) is compact. Local finiteness therefore leaves only finitely many smooth component intersections meeting these compact sets. For each such stratum C and each subset of variable hyperplanes in T, consider the incidence

It is smooth, since varying each hyperplane independently supplies its normal direction. Sard’s theorem for the projection to the hyperplane parameter space gives transversality for a general tuple; a countable atlas handles any noncompact stratum. Choose a tuple satisfying all the finitely many conditions and also the open conditions that it forms a coordinate basis and that none of its hyperplanes contains t_*. Use those coordinates for

Its derivative at a point has image the tangent space to the intersection of exactly the coordinate hyperplanes corresponding to vanishing coordinates, or is invertible when none vanish. The chosen transversality therefore gives, at h(z)=\psi(w),

for every smooth SNC stratum. The map is unbranched over t_*.

In local coordinates t_i on U, extend the coordinate functions of h holomorphically off H to functions G_i on local opens of \widehat Z. In \widehat Z\times U', the equations \psi_i(w)-G_i=0 have surjective differential on each H-stratum by (10.17). They cut a smooth submanifold of codimension b transverse to the SNC divisor. The cut H^\flat=H\times_U U' is therefore reduced SNC of pure dimension \dim V-1, and has codimension b+1 in the product ambient. Complete-intersection adjunction gives (10.16). In coordinates its last factor has the wedge frame dw/dt; this means a frame of \omega_{U'}\otimes\psi^*\omega_U^{-1}, not an inverse Jacobian.

The graph \Xi is a compact closed analytic subspace. Its actual line identity is

Lemma 10.1 gives a root P_* of \operatorname{pr}_V^*L on a neighborhood of \Xi, identified with \operatorname{pr}_{T'}^*R along \Xi. Form the same two full normalized covers there. For the second cover use coefficients of s in meromorphic canonical generators from the first factor V. This defines its relative tautological form without requiring a Cartier relative canonical line on the singular cover; changing to a generator after differential pullback changes the coefficient by a qth power and identifies the total root algebra and its normalization. Resolve and principalize the full tautological ideal functorially on this neighborhood, before cutting by the graph.

Near w_0, choose a frame \bar p of R whose rth power is \psi^*\ell for a downstairs frame \ell; a local root of a unit gives such a choice. Along the compact graph slice, compare the pullback of P_1 with P_* by sending the boundary frame p_1 to \bar p. Lemma 10.1 extends this to an ambient root isomorphism as a germ. On the graph, its ratio with the prescribed comparison is a \mu_r-section equal to one on the compact slice, hence equal to one on an open graph neighborhood of that slice. Properness of \Xi\to T' permits a base shrink on which the ambient comparison is defined and agrees with the prescribed one along the entire graph: remove the closed image of the complement of that neighborhood. This argument also applies to a nonreduced graph. The comparison identifies the first cover with the local ordinary product cover, and product normality identifies the second cover as well. Apply the same fixed smooth-functorial principalization to the same full ideal and ordered boundary data on both sides; functoriality identifies the results. These are isomorphisms of resolved ambient germs over the product base preserving the whole divisor ideal, not only isomorphisms of their supports.

Let H' be the graph cut inside the reduced inverse section divisor on this global resolution. Locally it is exactly H^\flat above, so it is reduced SNC with the asserted pure dimension. It is compact, being over the compact \Xi by proper maps. The ambient projection is a submersion near it by the local product comparison. The transverse cuts of the globally smooth inverse-divisor components are smooth; split them into their disjoint connected components. Compactness and local finiteness leave only finitely many. Their closed intersections are smooth and compact, hence proper over T'. Finally, the relative metric construction above, applied over the Kähler product neighborhood of \Xi, gives one Kähler form near H'. Its restrictions supply all the required relative Kähler forms.□

Proof. We use Saito’s analytic category of pure real Hodge modules [46, Sections 1.4 and 1.8], which permits discrete real indices for the V-filtration. The pure properties supplied above are in that category. If the constant-source theorem is read with the stronger rationally indexed pure conditions, those imply the broad real specializability conditions, and induction on support dimension gives the pure clauses of Definition 1.8 as well. On a smooth dense analytic Zariski open Z^\circ of Z, the piece Q gives a polarized real variation \mathbb H of weight w-\dim Z, by Lemmas 1.9 and 1.13 of that source. Theorem 16.2.1 and Corollary 16.3.6 of [45] give a pure object B\in\mathrm{pHM}_Z(T,\R,w) extending this same variation with its real polarization. Use its prime holomorphic filtered component.

Both underlying complex regular holonomic modules are the intermediate extension \operatorname{IC}_Z(\mathbb H_\C). For the Saito filtered module use strict support and regularity in Remark 1.11 of [46]; for the Sabbah–Schnell component use [45, Section 14.2.13 and Theorem 14.7.1]. Regular Riemann–Hilbert and uniqueness of intermediate extension [45, Theorem 13.2.11 and Section 13.2.12] give the unique unfiltered identity extending the chosen identity on Z^\circ. It remains to prove that this identity preserves F at the boundary.

On a smooth support U of dimension e, write \mathscr H for the flat holomorphic bundle of the variation, with decreasing Hodge filtration. Its right filtration is

Indeed the left filtration is F_p^L=F^{-p}, and the side change is F_p^R=\omega_U\otimes F_{p+e}^L. The same formula is recorded in [45, Section 16.3.12]. The right closed transfer (11.7) introduces no further shift. Thus the two right filtrations agree away from the boundary of Z^\circ.

Work locally on T. Choose a holomorphic g whose zero set contains that boundary but no local component of Z, and split into the finitely many local strict-support factors if necessary. Such a g exists by prime avoidance: the boundary has smaller dimension than each local support branch. If the boundary is empty there is nothing to prove. Use the graph embedding in T\times\C_t, and let Q_g denote the transferred module. Put b=\dim T. Saito’s left convention uses increasing F^L and decreasing V_L^\alpha with t\partial_t-\alpha nilpotent on its \alpha-grade. On the graph ambient the conversion is

The second rule follows because transposition sends t\partial_t to -t\partial_t-1. In particular V_L^{>-1} becomes V^R_{<0}. The sign change of a normal \partial_t does not change a generated sum.

We verify the filtered surjectivity needed in Saito’s reconstruction. Put \Psi=\psi_{g,1}Q and \Phi=\phi_{g,1}Q. Strict support gives a surjective \mathrm{can}:\Psi\to\Phi and an injective \mathrm{Var}, by [46, Proposition 1.5]. Forgetting F, this is the nilpotent middle-extension monodromy quiver of holonomic modules. Its monodromy filtrations centered at zero obey

by [45, Lemma 3.3.13]. The centers of the weight filtrations in Saito’s pure definition are w-1 for \Psi and w for \Phi. Thus (11.19) says \mathrm{can}(W_i\Psi)=W_i\Phi. Condition (1.8.2) of Definition 1.8 and the direct-factor Lemma 1.15 of [46] put both \gr_i^W\Psi and \gr_i^W\Phi in Saito’s pure real category of weight i. The induced surjection on each grade is a filtered pure morphism, hence is F-strict by its Proposition 1.10.

At a stalk, take y\in F_p\Phi and choose i with y\in W_i\Phi. Strictness on the ith grade lifts its class by an element of F_p\Psi\cap W_i\Psi. Subtract its image and repeat on the next lower weight. The finite monodromy filtration terminates this procedure. It proves

The finite lifting argument obtains filtered surjectivity from pure strictness on each weight grade.

Saito’s filtered middle-extension formula [46, equation (1.7.2)] now applies: the local pure factor satisfies its specializability conditions, the zero set of g does not contain its support, and \mathrm{can} is filtered surjective. After (11.18), that formula is

where j_{\mathrm{op}}:T\times\C^*\hookrightarrow T\times\C and the intersection is inside (j_{\mathrm{op}})_*j_{\mathrm{op}}^{-1}Q_g^R. The strict-support module and its V-pieces embed there by restriction.

For the Sabbah–Schnell component, strict real specializability, pure support, and strictness of \mathrm{can} follow respectively from Definition 14.2.2, Theorem 14.2.19, and Corollary 14.2.23 of [45]. Its unfiltered intermediate-extension property makes \mathrm{can} surjective, so it is a filtered middle extension. Definition 10.6.1, Remark 10.6.2, and Proposition 10.6.5 of the same source give precisely (11.20) for its right component: for \alpha<0, the filtered V_\alpha is the intersection with the filtration off the divisor, and the full module is generated from V_{<0} by normal derivatives.

The unfiltered intermediate-extension identity preserves the canonical V-filtration and, by (11.17), preserves F off the divisor. The two right sides of (11.20) are therefore identical. This proves equality of F on the graph. Formula (11.7) recovers the module before graph transfer, and its filtration, as the sections annihilated by the normal ideal. Hence the identity is filtered on T. The local identities glue by uniqueness of the unfiltered identity. The comparison uses the actual generic variation and its filtration; the two theories’ internal twist symbols need not have the same weight convention.

A pure object of Sabbah–Schnell is an object of \mathrm{WHM}(T) with its one-step weight filtration, by Section 14.2.14 of [45]. Its Theorem 16.3.10, on the smooth projective T, gives negative-ample graded de Rham vanishing for either filtered component. Sections 8.4.1–8.4.9 of that source identify the perverse right de Rham convention with the Spencer complex ending in the module in degree zero, with its filtration grading preserved. The theorem therefore gives (11.16) for every homogeneous degree p in our convention. This proves the lemma.□

Proof. On compact T the locally finite strict-support decomposition of each pure quotient of Proposition 11.1 is finite. Apply Lemma 11.2 to each factor. The finite filtration of M is F-strict, so its short exact sequences remain exact after \gr^F\DR_T. Their hypercohomology long exact sequences give (11.16) for M itself. This deduction uses the comparison for the pure quotients and the strict filtration of M.

By (11.2), the p=1 right graded de Rham complex has exactly two terms:

Since N is invertible, its tensor is exact, and negative hypercohomology gives

The two-term calculation uses no local freeness of \ker\sigma, so it applies to torsion as well.□

Proof. Use (10.9) together with

All are valid for negative j because I is invertible. Write \partial_{j,k}:g_*\mathcal T_{j,k}\to R^1g_*\mathcal A_{j+k} for the connecting map of the first sequence. The shorter orders make g_*\mathcal T_{j,k}\to g_*\mathcal A_j surjective. For k>1, its kernel is the image of g_*\mathcal T_{j+1,k-1}, by left exactness in the second sequence. The shorter order k-1 in degree j+1 lifts that image through g_*\mathcal T_{j+1,k}, which maps into g_*\mathcal T_{j,k+1}. The connecting map vanishes on the kernel. For k=1, the leading map is the identity and its kernel is zero. This proves the unique factorization (12.1). If it is zero, exactness of the first sequence gives the desired epimorphism.

Here is the derivation calculation on germs. A leading section in degree j lifts to a section of \mathcal T_{j,k} after a base shrink. Choose local representatives x_\alpha\in I^j on an ambient open cover near E. Their differences belong to I^{j+k}, and their classes modulo I^{j+k+1} represent \delta_k(x). For a second section z of degree l, take simultaneous representatives. On an overlap,

The last term lies in I^{j+l+2k}\subset I^{j+l+k+1}. The first two terms reduce to multiplication by the leading coefficients. Their Čech classes therefore give

This uses the ordinary multiplication action of layer sections on their first cohomology, and gives the asserted graded derivation.

For the chain rule, choose simultaneous shorter representatives G_{i,\alpha}\in\OO_Z of the degree-zero sections t_i. Their differences lie in I^k. Shrink the local opens so that F(G_{1,\alpha},\ldots,G_{b,\alpha}) is defined. Its holomorphic Taylor difference, modulo the square of the differences, is \sum_iF_{t_i}(G_\alpha)(G_{i,\beta}-G_{i,\alpha}). The square lies in I^{2k}\subset I^{k+1}, and the coefficient restricts to F_{t_i}(t) on E. This gives (12.2) before any restriction to a generic parameter. It is valid when the target contains torsion. The connecting map and all representative calculations are natural for restrictions and the prescribed ambient root isomorphisms.□

Proof. All statements are on germs, so choose a common parameter shrink and simultaneous shorter lifts of the finitely many sections x,t_i. Choose local ambient representatives near E, indexed by \alpha, with

They exist by the smaller orders in Lemma 12.1; for k=1 the shorter truncation is already the layer. Pull them to \widehat Z, suppressing pullback symbols, and put

On H the G_{i,\alpha} restrict to the coordinate functions of h. The Q_i’s and a reduced equation of H are therefore the regular graph-support equations. The logarithmic insertion and D_E=\rho^*E\geq H give the precise pole bounds

The last inclusion uses k\geq1. Thus the quadratic and cross difference terms have no pole on the first-factor support H.

In algebraic local cohomology consider the local sections

The H-pole is already in \eta_\alpha. These are finite-pole generalized fractions with a fixed ordering of the graph equations. To expand their differences, first quotient the localization along H by forms regular there, and then localize for the Q_i’s. Each polar class is killed by a power of a reduced equation of H, and every G_{i,\beta}-G_{i,\alpha} is a multiple of that equation. The change from Q_{i,\alpha} to Q_{i,\beta}=Q_{i,\alpha}-(G_{i,\beta}-G_{i,\alpha}) therefore has a finite geometric expansion on each polar class. By (12.9) only its linear terms survive. This proves the exact generalized-fraction equality on an overlap

There is no infinite series or division by a Jacobian in this identity.

Let C be the first Čech cochain in (12.10). Let E_i have the numerator of its ith summand but only the denominator P_\alpha, and let E_i^{(l)} denote that cochain with the additional denominator Q_{l,\alpha}. The simple fractions C,E_i are cocycles: their reductions are respectively the cocycles for \delta_k(x) and x\delta_k(t_i); on a triple overlap the possible change of the representative of x is a cross term with no H-pole by (12.9). The description of the residue insertion on representatives and complete-intersection adjunction identify their classes with \widetilde c,\widetilde e_i in F_0M^\flat. More explicitly, the residue of a log form \eta on H, pulled with dw/dt, is represented by [\eta\wedge dw/P_\alpha]; the determinant of the graph conormal gives exactly that frame. Thus this identification holds also at ramification. The cochains E_i^{(l)} need not individually be cocycles.

The numerator of E_i is independent of w. The right canonical action is minus differentiation of the coefficient of the volume form, so at the cochain level

Write \check d for the Čech differential. Equation (12.10) is \check dS=C+\sum_iE_i^{(i)}. Multiply it by \det J and use JJ^\#=(\det J)\id together with (12.11). The result, still at the cochain level, is

The placement of J^\#_{ji} after the right derivative is part of this exact formula.

These cochains lie in the end term of the relative right de Rham complex for the product projection, with first-factor coordinates held fixed. That term has no outgoing relative differential, so a Čech coboundary there is a total coboundary in direct image. The action in (12.11) induces the base right action there. Passing (12.12) to M^\flat gives (12.5). This uses the natural map from these Čech classes to direct image, and does not require that all direct-image classes be computed by this cover.

The term \widetilde c\det J is in F_0. Moving a holomorphic function past a right vector field changes an operator by order zero. Taking \gr^F_1 of (12.5) gives (12.6). With unchanged coordinates J=\id, the exact formula itself gives (12.7). If b=0, there are no Q_i; the undivided difference of the \eta_\alpha in the support module of H is the cochain for c(x) and is a total coboundary. This proves the last assertion as well.□

Proof of Proposition 10.2. Assume the shorter orders and use Lemma 12.1. First suppose b>0, and fix an arbitrary point t_*\in T=\PP^b. Proposition 10.5 gives the coordinate-power map \psi:T'=\PP^b\to T, the line R=\OO_{T'}(1) with R^r\simeq\psi^*\OO_T(1), and a single compact SNC graph support H'\subset\mathscr W with submersive projection p':\mathscr W\to T'. It is unbranched over t_*. Put

The global graph proposition verifies every hypothesis of Proposition 11.1: the support has pure dimension \dim V-1, globally smooth finitely many components, and proper smooth closed intersections with one ambient Kähler form. Thus F_0M'=R^1h'_*\omega_{H'}\hookrightarrow M', in the absolute canonical convention, and Corollary 11.3 applies.

Take the local leading section x=u^{-m}. On a local graph chart choose a frame \bar p of R with \bar p^r=\psi^*\ell for a downstairs frame \ell. The local formula

defines a global morphism

We verify the transition, including on the branch locus.

Let t'=t'(t) and w'=w'(w) be changes of coordinates, and put A_t=\partial t'/\partial t, B_w=\partial w'/\partial w. These matrices are invertible. If \bar p'=\lambda\bar p, then \lambda^r=\psi^*\mu for the downstairs unit \mu changing the frame of \OO_T(1). Locally even at a branch point, \lambda descends as a holomorphic unit: take an rth root of \mu on a small downstairs neighborhood, and observe that the remaining ratio has rth power one and hence is locally constant. Denote the resulting downstairs unit also by \lambda.

Compare the local root pairs by sending their new boundary frame to \lambda p_1. Lemma 10.1 extends this power-compatible comparison to the ambient germs. Its pullback agrees with the transition of the global root near the graph, by the uniqueness and properness argument in Proposition 10.5. It therefore identifies both full covers, their tautological forms, and the fixed functorial principalizations. Under this identification u'^{-m}=\lambda^m u^{-m}. In the definition of e_i(x), the factor x multiplies outside the derivation. Hence no derivative of \lambda occurs. Naturality and the chain rule (12.2) transform the downstairs column of e_i’s by \lambda^m A_t.

The absolute graph factor changes by dw'/dt'=(\det B_w/\det A_t)\,dw/dt. Consequently, for the pulled columns and Jacobians,

The adjugate identity holds for every J: it follows for invertible J from the inverse formula and then for all J as a polynomial identity. Only the coordinate changes A_t,B_w are inverted. Since the vector basis changes by B_w^{-1}, the last equality is exactly the transition for the image of the frame \bar p^m in (12.13). This proves (12.14) on all of T'. At ramification the classes are the natural Čech pullbacks under the resolved ambient germ comparisons; a flat base-change isomorphism for R^1h_*\omega_H is not used.

By Lemma 12.2, the image of (12.14) lies in \ker\sigma. The line R^m=\OO_{\PP^b}(a+k) is ample because a+k>0. Corollary 11.3 makes the global map zero. At a point above t_*, the map \psi is locally biholomorphic and J^\# is invertible. The ambient product comparison then identifies the graph family and its Čech classes with the downstairs family. Thus the zero map implies e_i(u^{-m})=0 as a downstairs germ near t_*, including a class supported at that parameter. The point t_* was arbitrary, and the root comparisons allow any chart at it. It follows that these classes vanish locally everywhere.

The injection \tau_* and the invertible layer frame u^{-m} now imply \delta_k(t_i)=0 on every chart: multiplication by that frame is an isomorphism from the layer \mathcal A_k to \mathcal A_{-a}, including on first direct images. For any local x of degree -m, all e_i(x) vanish. The exact unchanged coordinate identity (12.7) gives c(x)=0 in M^\flat. The injection (11.2) brings this back to R^1h_*\omega_H, and the residue injection gives \delta_k(x)=0. When b=0, Lemma 12.2 gives c(x)=0 directly, and the same two injections give this conclusion; there are no parameter coordinates to treat.

In particular \delta_k(u^{-m})=0. The derivation rule for the invertible Laurent frame gives

so \delta_k(u)=0. For any local F\in g_*\OO_E, the section Fu^{-m} also has degree -m, and

Thus \delta_k(F)=0. Every local graded section is Fu^j, so \delta_k is zero in every integer degree. Lemma 12.1 closes the current order. Induction from k=1 proves Proposition 10.2.□

Completion of Theorem 1.2. Define global divisorial subsheaves of meromorphic functions on the normal space V by

They agree in each root chart with (\pi_*I^j)^{\mu_r}. Indeed, at a prime over S_i, an invariant meromorphic function of downstairs order v_i has order (r/d_i)v_i. Membership in I^j is equivalent to

At all other primes it is regular. The full invariant meromorphic algebra is the downstairs algebra, and normality extends the codimension-one test. Finite coherent pushforward and averaging by \mu_r are exact. This also proves coherence of the divisorial sheaves in (12.17) and of their quotients locally, and their valuation descriptions glue globally.

Since 0<d_i\leq r, the sheaf J_1=\OO_V(-S) is the ideal of the reduced divisor S. The same coefficient inequality gives J_1J_j\subset J_{j+1} for every j. Hence

is a coherent \OO_S-module, and \mathcal B_0=\OO_S. Taking invariants of Proposition 10.2 preserves its epimorphisms: lift an invariant local section upstairs and average the lift. For l<n, define the sheaf of complex vector spaces

Set also \mathscr F_{n,n}=0. The quotients J_l/J_n are considered on the underlying space of S, just as before. The invariant lifting surjections give exact sequences of complex vector space sheaves

In particular \mathscr F_{l,n} has a finite filtration with the full quotients F_j, l\leq j<n. Each F_j is a coherent \OO_T-module by properness of f; coherence as an \OO_T-module is not asserted for \mathscr F_{l,n}.

Cartier translation by G in (12.17), together with the actual line identity \OO_V(G)=L defined by s, gives

Projection formula and (10.2) consequently give

This is an identity of the actual holomorphic lines and sheaves.

For N>0, each -Nr\leq j<0 is uniquely j=l-tr with 0\leq l<r and 1\leq t\leq N. By (12.19), F_j=F_l(t). Analytic GAGA and Serre’s coherent finiteness and vanishing theorems [50, Section 3, no. 12, Theorems 1 and 3], [49, Section 66, Theorems 1 and 2] imply, for every i>0,

These constants are independent of N, and they are zero for i>\dim T. The long exact sequences of the finite filtration (12.18) give the same bounds for h^i(T,\mathscr F_{-Nr,0}), as well as finiteness and vanishing above that fixed dimension. Serre vanishing is being applied only to the coherent quotients F_l(t).

There is at least one section of F_0(t) for every t\geq1. Choose a point of the nonempty S and a section of \OO_T(t) nonzero at its image; its pullback is nonzero there. Thus \sum_{j=-Nr}^{-1}h^0(T,F_j)\geq N. Additivity of the finite Euler characteristics in (12.18), and the uniform higher bounds for both the quotients and \mathscr F, give a constant C independent of N with

For example one can take C=2\sum_{i=1}^{\dim T}C_i.

Finally, J_{-Nr}=\OO_V(NG) and J_0=\OO_V. Degree-zero direct image on the underlying support gives

The exact sequence on V loses at most the fixed finite dimension h^1(V,\OO_V) when lifting these quotient sections to H^0(V,\OO_V(NG)); finiteness follows from proper coherent direct image to a point. Equation (12.21) therefore makes these section spaces unbounded. For some N the line \OO_V(NG) has two independent sections. Their quotient is a nonconstant meromorphic function on the irreducible normal V, so its Iitaka dimension is at least one. Since \OO_V(G)=L=\OO_V(qA) is the actual adjoint identity, this proves \kappa(V,A)\geq1, as claimed.□

Proof of Theorem 1.1. If \kappa(X,D)\geq1, Proposition 5.2 gives semiampleness on the original X, by descent of the generated actual Cartier multiple. It remains to treat \kappa(X,D)=0.

Choose a positive integer m_0 for which the actual line \mathscr L=\OO_X(m_0D) is invertible and has a nonzero section s_0. A connected normal space is irreducible, so this section is locally a nonzerodivisor. Let

It is the normalized effective rational \Q-Cartier divisor of the plurisection, and the chosen line and section give the actual equivalence M\sim_\Q D.

Suppose M\ne0. Proposition 5.1 produces a normal ordinary \Q-factorial compact Kähler dlt fourfold (V,B) with effective rational boundary, analytically nef actual adjoint A=K_V+B, and a nonzero effective rational \Q-Cartier divisor P such that

The space V is irreducible, being bimeromorphic to X, and has the special projective resolution in Lemma 5.6. Theorem 6.1 therefore makes the actual restricted adjoint semiample on the entire reduced floor S=\floor B. This floor is nonempty because P\ne0. Every hypothesis of Theorem 1.2 is now satisfied, so it gives \kappa(V,A)\geq1, a contradiction.

Thus M=0. The section s_0 is nowhere zero. Indeed, if a local representative were a nonunit at a point of the normal space, a minimal prime over its nonzero principal ideal would have height one, and would give a component of its zero divisor. Consequently s_0 supplies an isomorphism of holomorphic lines

This proves the asserted actual \Q-linear triviality in Iitaka dimension zero, and in particular semiampleness there. Together with the positive-Iitaka-dimensional case it proves the theorem.□