Lifting sections from the reduced support of an adjoint

1 Introduction

A nonzero section of an adjoint divisor gives an effective representative; abundance requires enough sections to define a morphism. Adjunction suggests looking first on the support of the zero divisor, where a multiple may already be generated. We prove that, when this support lies in the coefficient-one boundary, generation on the entire reduced support forces positive ambient Iitaka dimension.

The restriction is a line bundle on the whole reduced scheme S=G_{\mathrm{red}}. Sections on its separate components must agree along their intersections. The support inclusion may be strict: in C=S+H+C_0, the extra coefficient-one divisor H may meet S and any of its strata, while C_0 has coefficients less than one. No additional nefness hypothesis is needed. Curves in S have nonnegative G-degree by semiampleness, and curves outside S have nonnegative degree by effectivity. The rational coefficients of C may be arbitrary in [0,1].

The first application separates abundance from the task of obtaining a first section.

The proof uses lower-dimensional abundance and ordinary minimal models for effective log canonical pairs. Theorem 1.2 is independent of the existence of a first section in dimension four. It applies to arbitrary rational boundary coefficients and supplies a qualitative generated Cartier multiple, with no uniform index. Section 7 transfers the same conclusion to every algebraically closed field of characteristic zero.

The same supported argument gives a reduction in all dimensions.

The premise concerns pseudo-effective canonical divisors and includes those that are not nef. Hashizume’s reduction supplies nonvanishing and ordinary log minimal models under this premise; it does not supply the semiampleness proved here.

A separate companion provides the first section needed in dimension four.

This resolves rational log abundance positively in the stated range, using Theorem 1.2 and the nonvanishing result in OpenAI, Fourfold nonvanishing by minimal metrics and moving jets (September 27, 2026), Corollary 1.2 [27]. The proof of the supported theorem and of abundance after nonvanishing uses no result from that companion.

Boundary deformation and extension

The relation between boundary geometry and ambient sections already appears in dimension three. Kawamata’s alternative proof of Miyaoka’s numerical-dimension-one theorem constructs compatible infinitesimal boundary deformations using finite covers and Hodge theory, and then produces a moving family [16, Section 4]. The question of how infinitesimal boundary data yield ambient positivity also motivates the lifting problem considered here.

Log abundance through dimension three rests on the surface and threefold theorems, including the corrected Keel–Matsuki–McKernan argument [17, 18, 11] and semi-log-canonical gluing [10]. Liu–Xu proved abundance after nonvanishing through dimension five under the additional numerical-dimension bound \nu\leq1, together with a corresponding conditional reduction from smooth canonical nonvanishing in dimension d-1 and in numerical dimension one in dimension d [25, Theorems 5.1 and 5.4]. Theorem 1.2 removes that numerical-dimension bound in dimension at most four; the lifting theorem itself holds in every dimension.

On the analytic side, Demailly–Hacon–Păun proved extension for plt pairs without requiring an ample boundary part and explained its connection with nonvanishing and good minimal models [8, Theorem 1.7 and Corollary 1.8]. Their nef plt corollary gives restriction surjectivity in the presence of a suitable effective supported representative. Chan–Choi subsequently removed one support-containment restriction in the log-smooth plt setting [6, Theorem 1.6.1]. Theorem 1.1 starts with semiampleness on an entire reducible support and allows additional coefficient-one components to meet it. Compatibility at those intersections, and obstructions concentrated on smaller strata, are central to its proof.

The passage to abundance also needs distinct results about gluing and minimal models. Birkar’s effective log minimal model results provide the birational models used in the induction [3, 4]. Fujino–Gongyo show that semiampleness on the normalization of a semi-log-canonical pair, with its conductor boundary, descends to the original pair [12, Theorem 4.3]. Their nef log-abundance theorem requires abundance on the ambient variety and on the normalizations of all log canonical centers [12, Theorem 4.2]. Both requirements enter the proof of Theorem 1.2.

The obstruction on a finite neighborhood

After replacing G by a multiple, its generated restriction gives f:S\to\mathbb P^\ell with \mathcal O_V(G)|_S\simeq f^*\mathcal O(1). We want to extend functions and sections through successive infinitesimal neighborhoods of S. There is an immediate difficulty: f itself need not extend to any neighborhood. The proof must therefore measure the lifting obstruction using only the morphism on the reduced boundary.

Pass to the bundle of nonzero frames of \mathcal O_V(G), where the pulled-back line bundle has a tautological trivialization. Scalar multiplication of a frame gives an action of \mathbb G_m. A function or form has weight a if this action multiplies it by the character \lambda^a. Two cyclic-cover constructions, retaining all components of each normalized cover, produce a reduced Cartier divisor E=(y=0). After an equivariant resolution there is a logarithmic top form \sigma whose residue along the reduced resolved boundary has positive weight. Poles over the extra boundary H remain present, including above its intersections with S.

Locally, the finite neighborhoods are defined by (y^k). At the first stage where a lift from y^k=0 to y^{k+1}=0 is not yet known, differences of local lifts are multiples of y^k. Their coefficients form a cohomology class on E: this is the obstruction. Because products of two such errors vanish at that order, the connecting map on functions satisfies the Leibniz rule. Thus the obstruction is a derivation. A suitable character part of the residue map embeds its cohomology group into the cohomology of logarithmic residues.

The identity \mathcal O_V(G)|_S=f^*\mathcal O(1) lifts f to a map between the bundles of nonzero frames. On a chart trivializing \mathcal O(1), the target of this lifted map has projective-base coordinates and an invertible frame scalar. Changing the base coordinates gives a horizontal direction; rescaling that scalar gives the vertical direction. The proof first eliminates the horizontal part of the obstruction, then uses the nonzero weight of the frame action to eliminate the vertical part. This distinction explains why positivity of the residue weight is useful.

To retain classes supported on singular fibers or smaller strata, the residue comparison is made in a filtered direct image of a Hodge module on the graph of the boundary morphism. Its lowest filtration term is the whole coherent residue cohomology sheaf. Its first symbol records the effect of differentiating in coordinates on the target of the lifted map. Saito’s strictness and Kodaira–Saito vanishing provide the required control of these filtered objects [30, 31, 28]; restricting to the smooth locus of a variation of Hodge structure would discard some of the obstructions we need to test.

A local calculation on the graph expresses the derivation and this first symbol in a single residue identity. The symbol has positive frame weight. If its horizontal part were nonzero, a finite cover transverse to that part, together with the adjugate of its differential, would produce a nonzero map from an ample line bundle into the kernel of the first symbol. Kodaira–Saito vanishing excludes that map. The remaining vertical term is the actual Euler derivative of frame scaling. The residue identity now says that a nonzero scalar, determined by its weight, times the vertical obstruction is zero. Hence that obstruction vanishes as well. Applying the same identity to arbitrary boundary functions removes the complete obstruction. This proves lifting at every finite order.

Taking finite-cover invariants and the degree-zero part of the frame action descends the lifted layers to divisorial sheaves on V. These layers recur with positive twists of \mathcal O_{\mathbb P^\ell}(1). Their accumulated sections grow while higher cohomology stays bounded; the loss in passing from a finite neighborhood to ambient sections is bounded by a fixed cohomology group. Hence the section spaces of multiples of G are unbounded, proving positive Iitaka dimension. If the boundary morphism has zero-dimensional image, the growing number of layers supplies this increase. The argument uses finite filtrations and does not require convergence of a formal neighborhood.

From lifting to abundance and other boundary methods

For Theorem 1.2, lower-dimensional abundance and conductor gluing generate the adjoint on the whole reduced floor. Theorem 1.1 then excludes a nonzero effective representative in the case \kappa=0. For positive Iitaka dimension, effective minimal models handle the generally non-nef adjoints on the Iitaka fibers. Numerical triviality there gives abundance, including on log canonical centers, and the preceding semiampleness and descent results apply.

The complex lifting argument also works under a weaker formulation: the canonical and boundary divisors are individually \mathbb Q-Cartier, the pair is log canonical, and it is klt off the reduced floor. This formulation permits transfer to other algebraically closed fields of characteristic zero. One descends finitely many defining data to a countable algebraically closed field embeddable in \mathbb C and transfers sections and generation by base change of the original line bundle and its evaluation map. No preservation of \mathbb Q-factoriality under that descent is needed.

A further method illuminates the boundary obstruction with different hypotheses. A complete conormal–period argument proves that a smooth projective complex fourfold with K_X nef and \kappa(K_X)=0 cannot have \nu(K_X)=2, without an Euler-characteristic hypothesis. Its period equation and trace tests at exceptional parameters kill successive obstructions; a quadratic count of invariant layers then forces growth. This method is not needed for Theorem 1.1.

Reading the proof

Section 2 constructs the two covers and a split residue insertion. Section 3 identifies the lowest filtered piece and proves the symbol-kernel vanishing. Section 4 separates horizontal symbols from the actual Euler action, and Section 5 uses these two conclusions to lift and count all finite layers. Section 6 proves the abundance implications over \C; Section 7 returns the resulting sections and generated multiples to the original field. The independent curve-base method is developed in Section A.

Conventions and the complex formulation

All boundaries are effective with rational coefficients in [0,1]. An adjoint is an actual \Q-Cartier divisor, and \sim_{\Q} denotes rational linear equivalence. A rational Cartier divisor is semiample if a positive Cartier multiple is generated by global sections; its Iitaka dimension is the dimension of the images of all sufficiently divisible nonempty complete systems, with value -\infty if no such system exists. For a nef divisor D on a projective d-fold,

for an ample divisor H; these intersections may be computed on a smooth resolution. We use log discrepancies, so coefficient-one divisors have log discrepancy zero. Standard conventions for lc, klt, dlt and dlt adjunction follow [19].

The complex proof uses only individual Cartier witnesses, rather than preservation of \Q-factoriality under a field extension.

This proposition is proved in Sections 2–5. Its hypotheses follow from those of Theorem 1.1 over \C, and its individual Cartier requirements are the finite witnesses used in the final field transfer.

2 Moving frames, cyclic covers, and residues

The first step in Proposition 1.5 is to embed the entire lifting obstruction into logarithmic residue cohomology. Moving to a frame bundle supplies compatible roots without choosing roots separately on open sets. The complete cyclic covers retain the character summand needed for an injective residue map, including the poles along the extra coefficient-one boundary.

Work over \C, and put A=K_V+C and S=G_{\mathrm{red}}. Replace (q,G) by (bq,bG), for a sufficiently divisible positive integer b, so that semiampleness on the entire reduced scheme S gives fixed identifications

Indeed, a globally generated power of the original restriction defines f by its complete linear system; surjectivity onto T is unnecessary. Further powers preserve this property. We also clear all Cartier indices and boundary denominators that occur below. Let s\in H^0(V,L) be the section with divisor G, and write

Here d_i>0, H is the reduced sum of the remaining coefficient-one components, and C_0 has coefficients in [0,1) and shares no component with S+H. Choose r divisible by q and all d_i, and put a=r/q. The singularity properties used in this section, and in the remainder of the supported-pair argument, are log canonicity and klt singularities away from S+H. A dlt pair has both properties. Apart from normality, the Cartier hypotheses used are that K_V and the indicated boundary components are \Q-Cartier, together with the actual identifications in (2.1).

2.1 The moving frame and the first cover

Let p:P=L^\times\to V and B=J^\times\to T denote the bundles of nonzero vectors. The tautological frame w trivializes p^*L on P. For \lambda\in\Gm, let \lambda multiply the vectors in both bundles by \lambda^r. All weights below are pullback weights: a tensor u has weight m if \lambda^*u=\lambda^m u. In particular w has weight r. The fixed restriction isomorphism gives

an equivariant projective morphism.

There is an invariant canonical comparison \omega_P\simeq p^*\omega_V, understood reflexively on the normal spaces. On a bundle chart with nonzero fiber coordinate t, it sends a top form \eta on V to p^*\eta\wedge dt/t. Replacing t by ut, with u invertible on the base, adds du/u to dt/t; its wedge with \eta vanishes. Thus the comparison glues and is invariant under scaling. The same construction applies to divisible adjoint powers. Moreover, the pullback pair (P,p^*C) is lc and is klt off S_P+p^*H: locally a log resolution is the product of a log resolution on V with \Gm, with the same discrepancy coefficients.

The ratio u=p^*s/w is a regular function on P. Form the whole finite algebra \OO_P[y]/(y^r-u) and let

be its normalization. This is the cyclic-cover construction of [9, Section 3.5], applied to the trivial line bundle; we retain its full-algebra convention. Throughout, normalization of a generically reduced algebra means normalization in its full total ring of fractions. Thus Z is a disjoint union of normal varieties if the generic algebra splits; no component is discarded. The polynomial is separable over every generic field, and normalization is finite since the schemes are of finite type over \C. The function y remains regular, and the action lifts by

It commutes with the deck action y\mapsto\zeta y, \zeta\in\mu_r.

2.2 The root of the adjoint tensor

The nowhere-vanishing frame w, the fixed isomorphism in (2.1), and the canonical comparison produce a meromorphic q-canonical tensor on P. As an adjoint section it has no zeros, so its divisor as a meromorphic tensor is -qp^*C. Its differential pullback to Z is a tensor \tau, nonzero on every component, satisfying

The canonical ramification formula used in (2.3) proves this equality, including its integral coefficients. Thus the construction depends on the given rational linear equivalence, not merely on a numerical equivalence.

Let \nu:\widetilde Z\to Z be the full normalized q-th root of \tau. Explicitly, for a meromorphic canonical frame \theta on a component with function field K, normalize in the finite reduced K-algebra

Changing \theta to v\theta changes c to c/v, so the algebras and the tautological form c\theta glue. Pullbacks of forms in this description are differential pullbacks. The deck group \mu_q acts by c\mapsto\zeta c, and the tautological form has its character \chi(\zeta)=\zeta. Even if the algebra splits, its invariant subalgebra is exactly K: the basis 1,c,\ldots,c^{q-1} has all the distinct characters. This observation will be needed for descent of ratios of forms.

The \Gm-action also lifts algebraically. If on a meromorphic chart \lambda^*\theta=j_\lambda\theta, using differentials relative to the parameter \lambda, prescribe

This respects the equation because aq=r and \lambda^*\tau=\lambda^r\tau; the factors j_\lambda satisfy the cocycle identity. These formulas define the action on the generic algebra and on its integral closure. More explicitly, normalization commutes with the smooth base extension by \Gm, and the isomorphism between the two pullback algebras over \Gm\times Z therefore induces an isomorphism of their normalizations. This gives a regular action, not just a separate lift of each scalar automorphism. It commutes with \mu_q, and the tautological form has weight a.

Take a projective (\Gm\times\mu_q)-equivariant log resolution of \widetilde Z and the inverse images of the boundary supports, and write its composite with \nu as

Functorial resolution and principalization in characteristic zero give this resolution by invariant centers and projective blowups [2, Theorem 1.1], [38, Theorem 1.0.1]. Indeed, functoriality for the smooth action and projection morphisms (\Gm\times\mu_q)\times\widetilde Z\to\widetilde Z lifts the action and its group identities at every step. We use the pulled-back tautological top form \sigma on Y; it has deck character \chi and weight a. We may choose Y smooth and quasi-projective with a \Gm-linearized ample line bundle. To see the latter point, compactify P as \PP_V(\OO_V\oplus L), with its scaling action. A sufficiently positive twist of the tautological bundle by an ample bundle from V is ample and linearized. Restrict to P and pull back through the finite covers, which preserves ampleness and linearization. For every invariant blowup center its ideal, and hence the tautological relatively ample bundle on the blowup, is linearized. Tensoring these with sufficiently high powers of the preceding ample bundles at successive steps gives the assertion for Y.

Define effective divisors on Y by

The union R+D_h is reduced simple normal crossing. Both R and \widehat E map scheme-theoretically to E: the function y vanishes on them. Write q_R:R\to E and q_{\widehat E}:\widehat E\to E for these projective maps, and set h=g\circ q_R:R\to B.

The spaces and morphisms are summarized in Figure 1. The form \sigma lives on Y; its residue will live on R and will be pushed through the top row to B.

The definition of D_h is essential here. An exceptional divisor over E\cap H_Z belongs to D_h and is omitted from R, but its possible simple pole is still allowed in (2.9). No disjointness condition on H and the strata of S has been used.

2.3 The residue summand

2.4 The obstruction to be detected

Put A_d=|g|_*(\OO_Z/I^d) for d\geq1, pushing the underlying sheaves of complex vector spaces in the Zariski topology along |g|:|E|\to|B|. This uses no morphism from a thickening to B. Fix k\geq1 and suppose that A_{d+1}\to A_d is surjective for 1\leq d<k. The current connecting map is \partial_k:A_k\to R^1g_*(I^k/I^{k+1}). Section 5 will show that it factors, after framing I^k/I^{k+1} by y^k, through a derivation

Here \widetilde F\in A_k is any local lift of F\in g_*\OO_E. The factorization and derivation rule will be proved there from the previous surjections. Since y has weight -1, this derivation has degree k for the frame action.

Restrict \delta_k to pullbacks of functions on B. In local coordinates t_i, the residue injection produces a global section of R^1h_*\omega_R(D_h)\otimes T_B, given by

The derivation rule makes this tensor independent of coordinates. The next two sections develop a differential-operator test for it. The graph identity that makes the test applicable, and that then annihilates \delta_k on all of g_*\OO_E, is proved with the induction in Section 5.

3 A filtered module on the boundary graph

The tensor v_k of Section 2.4 has coefficients in R^1h_*\omega_R(D_h). We construct a filtered right \mathcal D_B-module whose lowest piece is this entire coherent sheaf, so that differentiation in base coordinates gives a first symbol for the tensor. We then prove a vanishing statement for the kernel of that symbol on a projective base. Section 4 will apply it to the positive frame weight a+k.

Temporarily isolate the graph calculation from the covers. Let Y be a smooth complex quasi-projective variety, let R+D_h be a reduced simple normal crossing divisor with no common component between R and D_h, and let h:R\to B be a projective morphism to a smooth quasi-projective variety. Write b=\dim B, and let Q\subset Y\times B be its graph. The projection is not required to be proper, but its restriction to Q is projective:

Throughout this section D_h also denotes its pullback to the product.

3.1 Conventions and the lowest filtration piece

We use right \mathcal D-modules underlying graded-polarizable algebraic mixed Hodge modules with rational structures. On a smooth d-fold we twist the constant Hodge module by d, so that its underlying canonical sheaf starts at F_0, rather than F_{-d}. All subsequent localization and support functors carry this normalization. The required functorial statements are the localization and support operations, exterior products, smooth inverse images, and projective direct-image strictness in [30, 31, 32]; conventions are also explained in [36, 28].

Define

Here the brackets denote algebraic local cohomology, and the plus sign denotes differential-module direct image. Localization and cohomology with supports show that \mathcal M is the underlying filtered module of a mixed Hodge module. Its direct image has projective support, as above.

For a right module, the relative filtered de Rham complex for the projection has term

The tangent sheaf here is pulled back from Y. In particular its last term, in degree zero, is F_j\mathcal M.

The same calculation describes every filtration level locally: F_j\mathcal M consists of the graph fractions with simple initial denominators in the boundary and graph coordinates, and at most j extra denominator exponents in total, modulo the fractions regular in the R-support direction. We will use this explicit description across a ramified base change.

3.2 The consequence of Kodaira–Saito vanishing

For a mixed Hodge module \mathcal N_0 on a smooth projective variety T_0 and an ample line bundle H_0, Kodaira–Saito vanishing gives

We use perverse normalization, so the right de Rham complex ends in degree zero. The mixed version follows from the pure strict-support decomposition and the strict finite weight filtration; see [31, Proposition 2.33] and [28, Theorem 28]. Tate twists do not affect the assertion, which holds for every j.

4 Positive frame weight and the Euler direction

We now combine the graph module with the scaling action on the frame bundle. A positive weight has two consequences: after a transverse cover it gives an ample source for the horizontal symbol, and in the vertical direction it gives a nonzero Euler eigenvalue. The proof keeps these two operations separate until both have been identified on the same right differential module.

Return to Section 2. Thus T=\PP^\ell, J=\OO_T(1), and B=J^\times is the bundle of nonzero vectors, with projection p_B:B\to T. The parameter \lambda\in\Gm acts on B by multiplication by \lambda^r. Its action on Y preserves R and D_h, and h:R\to B is equivariant. All the modules in (3.1) inherit their natural geometric action.

Let \varepsilon be the infinitesimal generator on B. In a bundle chart with fiber coordinate \beta,

It is nowhere zero and spans the kernel of T_B\to p_B^*T_T. By weight m we mean pullback weight: \lambda^*s=\lambda^m s.

The first assertion concerns the symbol, and the second concerns the operator itself. Their combination will annihilate the obstruction. We prove them separately, keeping track of both canonical factors and filtration shifts.

The horizontal directions cannot in general be suppressed. For example, if V=\PP^2 and C is a smooth quartic, then A=K_V+C\sim\OO_{\PP^2}(1) and G=C\sim4A satisfy the supported hypotheses, while \OO_V(G)|_C is ample and nontrivial. The construction must therefore retain the full map f:S\to T.

4.1 A transverse power cover

Suppose first that \ell>0. Take the coordinate r-th power map of \PP^\ell and compose it with a general target automorphism, obtaining

It is finite flat. Put B'=(J')^\times, with its ordinary scalar action, and let p':B'\to T' be the projection. Raising frames to the r-th power gives a finite flat equivariant map \eta in

Along each nonzero fiber its differential is invertible.

Choose \phi transverse to the maps to T from every closed intersection of components of R+D_h involving a component of R. These intersections are smooth and there are finitely many of them. Here is the generic-transversality justification. For such an intersection L, the matching-pair space for the map L\to T and the translated power map is smooth: variation of the target automorphism supplies every target tangent direction. A general fiber over the automorphism group is smooth by generic smoothness in characteristic zero. This is exactly the required transversality. The finitely many conditions can be imposed simultaneously. Since the vertical differential in (4.3) is invertible, this also gives transversality to the corresponding maps L\to B. If a point of T is prescribed, one may in addition require that the branch divisor avoid it, an open condition on the target automorphism.

Let Q'=Q\times_B B'\subset Y\times B', and set

These are again mixed Hodge modules with projective support.

One can also express the transversality in this proof as a non-characteristic condition. The characteristic variety of the normal crossing graph module is contained in the union of conormals to the graph strata involving R. A characteristic covector killed by d(\id_Y\times\eta) has zero Y component; its remaining component annihilates both the stratum tangent image and d\eta(T_{B'}). Transversality forces it to be zero. Finite flatness also makes the higher \operatorname{Tor} groups of every graded filtration piece vanish. These are the two non-characteristic conditions of [30, Section 3.5.1 and Lemmas 3.5.3, 3.5.5]; in relative dimension zero they give the same filtration index and right canonical factor as the explicit calculation above.

4.2 The adjugate and the projective quotient

For vector bundles of equal rank the adjugate of d\eta defines a regular morphism

Indeed \det(d\eta) is a section of \omega_{B'/B}, so this is the usual determinant-twisted adjugate. It involves no division. If v has zero symbol, applying (4.6) to its tangent factor and then (4.5) gives a section

with zero symbol. In local matrices this is just d\eta\operatorname{adj}(d\eta)=\det(d\eta)\id. The construction preserves the weight of v.

Suppose the horizontal image of v is nonzero. Choose a point where that coherent-sheaf section is nonzero, and choose \phi as above with branch divisor avoiding its image in T. On an unbranched neighborhood the tangent map is an isomorphism preserving the vertical tangent line. Faithful flatness there shows that the horizontal image of v' is still nonzero.

Now take the diagonal quotient

It exists as an associated algebraic variety: a local section of B'\to T' identifies it with Y times that base chart, and changes of section glue these products through the action on Y. It is smooth. The linearized ample bundle on Y descends to a relatively ample bundle on this associated variety, so \overline X is quasi-projective. The invariant divisor D_h descends to a divisor \overline D_h, and the invariant closed support Q' descends to \overline Q.

The support map \overline Q\to T' is projective. Properness descends from the projective map Q'\to B', and a descended relatively ample bundle gives projectivity. The quotient changes both the ambient dimension and the support dimension by one; thus \overline Q still has codimension b+1. Define

Use the constant twist by \dim\overline X in this formula. These are algebraic mixed Hodge modules, and the second has projective support over the projective base T'.

Choose a frame of J' on U\subset T' and write a point of Y\times B'|_U as (z,t,u), with z\in Y, t\in U, and u\in\Gm. The coordinate change

turns the diagonal action (z,t,u)\mapsto(\lambda\!\cdot z,t,\lambda u) into the action on u alone. Since Q' and D_h are invariant, the support and divisor become products with \Gm as well. Localization and support commute with this product. Their normalizations give

To check the index, ordinary right smooth pullback of relative dimension one changes F_j to F_{j+1} and tensors with the relative canonical line. The normalized constant upstairs has one additional Tate twist. This cancels the index change: the normalized smooth inverse image is [1](1), and gives exactly (4.10). See [36, Section 30]. Relative direct image has the same relative dimension before and after this product, so its cohomological index stays one.

The relative canonical line in (4.10) has invariant frame du/u. This frame is intrinsic as a relative form: a change of bundle trivialization changes it only by a form from the base. In the product description the vertical symbol is zero and the horizontal symbol is the pullback of that on \overline{\mathcal N}. In particular F_{<0}\overline{\mathcal N}=0, by (4.5), (4.10) and faithful flatness.

Project v' horizontally and use (4.10). A homogeneous section of weight m on the nonzero vectors of J' corresponds to a section twisted by (J')^{-m}. Explicitly, if local frames satisfy e_j=g_{ij}e_i and a vector is u_ie_i, then u_j=g_{ij}^{-1}u_i; equality of expressions u_i^m s_i forces s_j=g_{ij}^m s_i, the transition rule for a homomorphism from (J')^m. Thus the projected zero-symbol section gives

It is nonzero if the original horizontal projection was nonzero. For m>0 its source is ample, contradicting Lemma 3.2. This proves assertion (i) of Proposition 4.1. For \ell=0 there is no horizontal tangent direction and assertion (i) is immediate.

4.3 The actual Euler action

It remains to prove assertion (ii), which does not follow merely from the vanishing of the vertical symbol. Work on a bundle chart of B and make the finite étale fiber cover

The generator (4.1) becomes u\partial_u. The coordinate change (4.9) again identifies the supported geometric construction with a product in the u direction. This is an identification of the underlying differential modules, with their natural equivariance, not just an identification of coherent filtration pieces.

A weight-m section is consequently u^m times a section independent of u, in the invariant relative frame du/u. Right action on top forms is negative Lie differentiation. Since du/u is invariant,

The product direct image retains this action, proving (4.2) on the étale cover and hence downstairs. This completes the proof of Proposition 4.1.

5 Infinitesimal lifting and section growth

We now apply the residue insertion and the two frame assertions to prove lifting through every finite order. Finite-cover invariants and the weight-zero part of the frame action will then return the lifted layers to V, where their positive twists force unbounded section spaces.

Retain the complex algebraic setup of the preceding sections. In particular, I=\OO_Z(-E)=y\OO_Z, the pullback weight of y is -1, and \Omega=\res(\sigma) has weight a=r/q>0. Write

for the injections of Lemmas 2.3 and 3.1. Thus \iota(e)=\Omega e, with pullback and residue understood. If F\in g_*\OO_E, the expression \Omega Fe means \iota(Fe): multiplication takes place on E, before the insertion. This convention does not require F to come from B.

For all integers j<d, we push the underlying sheaf of complex vector spaces of I^j/I^d along |g| using the Zariski-topology convention of Section 2.4; negative powers mean invertible fractional ideals. No morphism from an infinitesimal thickening to B is assumed. In contrast, multiplication by y^j identifies I^j/I^{j+1} with the coherent sheaf \OO_E, with an equivariant weight shift. Its additive higher direct images are therefore the underlying sheaves of the usual coherent higher direct images under the projective morphism g.

6 Abundance after nonvanishing

Over \C, the supported theorem closes the zero-Iitaka-dimension case once lower-dimensional abundance generates the whole reduced floor. For positive Iitaka dimension, the induction also needs effective minimal models of fiber adjoints, which need not be nef. We organize these two uses in one proposition, then verify its hypotheses in dimension four and under the all-dimensional premise.

Every effective representative below is effective up to actual \Q-linear equivalence. Ordinary log minimal models supply nef adjoints; semiampleness is the conclusion to be proved. The first lemma records exactly which section spaces survive passage to such a model.

6.1 Minimal models and the whole reduced floor

We may take the \Q-factorial dlt models in this lemma in the ordinary minimal-model existence statements below. A crepant dlt modification, when needed, has the same adjoint pullback and preserves nefness and all section spaces. The dlt modification theorem is, for example, [12, Theorem 2.8].

6.2 The induction step

6.3 The two applications

7 Transfer of the conclusions to characteristic-zero fields

The remaining task is to recover generation of an actual line bundle on the original variety over an arbitrary algebraically closed field of characteristic zero. We descend the specified divisor data to a countable algebraically closed field, embed that field into \C, and detect generation by the evaluation cokernel. Faithful flatness then transfers the same Cartier multiple back. The auxiliary minimal models used over \C play no role in this descent.

A A conormal–period argument in numerical dimension two

This section gives an independent use of the boundary: an alternative proof of the following restricted canonical statement.

Here \nu denotes the intersection-theoretic numerical dimension of a nef divisor. The effective divisor used below comes from the section supplied by \kappa=0. No condition on the irregularity or the Euler characteristic is imposed. The statement concerns the actual canonical divisor on a smooth variety; it is not a nonvanishing assertion for \kappa=-\infty or a statement for arbitrary klt pairs.

The proof has three parts. First, a dimension-preserving effective MMP puts the pluricanonical support on a reduced boundary whose adjoint defines a pencil. Second, root covers turn the obstruction to lifting through a nilpotent thickening into a period equation on that curve. Positivity of the source line kills its moving part; a separate trace test kills classes supported at exceptional parameters. Finally, finite invariant filtrations convert lifting into a quadratic lower bound for plurisections, contradicting \kappa=0.

The use of cyclic covers, infinitesimal neighborhoods and Hodge duality continues a method in Kawamata’s alternative proof of the numerical- dimension-one case for minimal threefolds [16, Section 4, Theorem 4.1 and Lemmas 4.2–4.3]. That result supplies the ancestry of this method, rather than the fourfold assertion of Theorem A.1.

A.1 The reduced boundary and a genuine pencil

Suppose that \nu(K_X)=2. A nonzero pluricanonical section gives 0\ne D_X\sim_{\mathbb Q}K_X, with D_X\ge0. On a log resolution p:Y\to X, let T be its reduced strict-transform support, let E_Y be the reduced exceptional divisor, and write K_Y=p^*K_X+F, with F\ge0 exceptional. Then

For some rational c>0, p^*D_X\le D_Y\le(1+c)p^*D_X+F+E_Y. Monotonicity, birational invariance and exceptional invariance of \kappa and Nakayama’s \kappa_\sigma therefore give dimensions 0 and 2 for this effective log adjoint [25, v1, Lemma 2.5 and proof of Lemma 4.1]. Effective fourfold minimal-model existence and termination with scaling give a projective \mathbb Q-factorial dlt model [24, Corollary D(i), Lemma 2.9(i)], [5, Theorem 4.1(ii),(iii)]:

Choose the scaling boundary to contain a fixed ample rational divisor. For a positive limiting scaling parameter, the dlt pair and this ampleness hypothesis allow the first termination clause; limit zero with every parameter positive uses the already obtained log minimal model and the second clause; an attained zero means the current adjoint is nef. The nonextracting MMP preserves the displayed support and both dimensions. The underlying V is klt.

The sheaves \mathcal O_V and \mathcal O_V(-B) are Cohen–Macaulay by the local vanishing theorem for integral \mathbb Q-Cartier divisors on a klt variety [22, Theorem 2 and Example 4.1]. Their quotient shows that B is a Cohen–Macaulay threefold. Adjunction on the whole union, including conductor gluing on its normalization, gives an slc pair (B,\Delta_B) with K_B+\Delta_B=A|_B. Threefold slc abundance makes this restriction semiample [10, Theorem 0.1]. Write D=\sum c_iB_i, c_i>0, and choose an ample divisor H. Then

All summands are nonnegative. The image of a generated multiple on each B_i consequently has dimension at most one, and one component has one-dimensional image. Two general linear forms with common center disjoint from this image generate the restricted line bundle everywhere, including components mapped to points. After fixing one divisible integer q, we obtain

where f:B\to\mathbb P^1 is surjective. Its fibers need not be connected. This is a morphism of the boundary; no ambient fibration is required.

A.2 Root covers and the two conormal frames

The pencil exists only on B. We now construct local covers on which its conormal line and absolute dualizing line have compatible frames. The construction must retain the entire reduced divisor, including components over individual parameter values.

Fix r divisible by q and every d_i, and put a=r/q>0. These integers will not vary with the infinitesimal order. Near a compact reduced fiber of f, choose a disk U and a frame \ell of \mathcal O(1)|_U. There is an analytic neighborhood W of B_U, a root P^r\simeq L|_W, and a boundary frame p with p^r=f^*\ell. Indeed choose a simultaneous triangulation compatible with the fiber and boundary [7, Theorem 1.10]. In successive barycentric subdivisions take the open stars of the compact fiber subcomplex. Their regular-neighborhood retractions preserve the boundary subcomplex. These stars are cofinal among neighborhoods of the fiber, giving arbitrarily small W for which W and W\cap B retract compatibly onto it. The retractions are the elementary regular-neighborhood consequence of the triangulation, not an additional assertion of the cited triangulation theorem. The Kummer obstruction in H^2(W,\mu_r) vanishes because L is trivial on that fiber. Twisting the root by a \mu_r-torsor, using the corresponding H^1 isomorphisms, gives the prescribed boundary frame. Properness then permits shrinking U.

Put Q=\omega_V(B) and M=Q\otimes P^{-a}, with reflexive powers understood; M^{[q]}\simeq\mathcal O_W. Normalize the full fiber product of the cyclic algebras

The first multiplication uses the section defining G, and the second uses this specified power trivialization. Retaining every component gives a finite cover \pi:Z\to W with group \mu_r\times\mu_q and quotient W. Its tautological section y has reduced Cartier zero divisor E; the ramification over B_i is r/d_i, and the second cover is unramified in codimension one. The two Hurwitz formulas are

The finite-map discrepancy rule gives klt Z and lc (Z,E) [23, Proposition 5.20]. In this setting it follows by exposing any divisorial valuation downstairs, normalizing its finite pullback, and using the DVR Hurwitz coefficient e-1: A_{\mathrm{up}}(F)=eA_{\mathrm{down}}(E_0) for a divisor F above E_0, where A denotes log discrepancy for the corresponding Hurwitz pairs. For the second displayed formula these are (Z,E) and (V,B). For the first they are Z with zero boundary and V with boundary \sum_i(1-d_i/r)B_i; the latter is klt because every coefficient of the reduced dlt boundary has been lowered. Thus the lc and klt inequalities hold for all divisorial valuations. The torsion root gives actual isomorphisms

Klt Cohen–Macaulayness makes Z Gorenstein; Cartier adjunction makes E Gorenstein. The normalized adjunction pair has conductor coefficient one at nodes and coefficient zero at regular codimension-one points, so E is slc [20, Lemma 11.16 and Theorem 11.17]. As a reduced union of lc centers it is Du Bois [21, Theorem 1.4].

The reduced families used here are algebraic, also through exceptional parameters. One explicit construction performs the two normalized cyclic constructions over the root gerbe \sqrt[r]{L/V}=[L^\times/\mathbb G_m], where \lambda acts by \lambda^r. An object over a map h:S\to V is a line bundle P_S together with an isomorphism P_S^r\simeq h^*L. Over B_U, the trivial line with power identification given by f^*\ell defines the selected morphism \sigma:B_U\to\sqrt[r]{L/V}|_{B_U}. Pull back along \sigma the reduced divisor of the already normalized cover. Thus the normalization precedes this restriction. The root-gerbe description is standard [1, Definition 2.2 and Proposition 2.3]; on the possibly singular V it follows from the same local line-bundle trivializations. The normalization and singularity comparisons used here are given by the following local construction. Locally the substitution v=w^r in the frame parameter identifies the construction with a fixed normalized cover times that parameter; normalization commutes with this étale localization. It does not require normalization to commute with restriction to B. A specified boundary root gives a finite étale section of the root gerbe: its pullback along any other root is the finite étale torsor of power-compatible isomorphisms. Pulling the reduced divisor back by this section preserves its total-space singularity properties. The resulting E_U\to U comes from a projective algebraic family with reduced pure three-dimensional Gorenstein Du Bois total space.

Globally take a finite cover \rho:T'\to\mathbb P^1 with a positive line R satisfying R^r=\rho^*\mathcal O(1); the degree-r power map with T'=\mathbb P^1 and R=\mathcal O(1) suffices. The same construction produces a reduced algebraic family g':E'\to T'^\circ on a nonempty open T'^\circ. Remove branch values, vertical-component images and the finitely many parameters needed for flatness, slc surface fibers, coherent base change and topological local systems. Keep the original families through the removed points. In what follows, a good disk, annulus or locus lies inside the specified open on which these flatness, base-change and local-system properties hold, or its corresponding open in a local chart family. An isomorphism between specified boundary roots extends uniquely as a germ near a compact fiber, since its isomorphism sheaf is a finite \mu_r-torsor and the retractions preserve its sections. These germ isomorphisms respect every actual ideal power and its connecting map. They give the cocycles comparing all local root choices with the one family on T'^\circ.

Set I=\mathcal O_Z(-E) and \mathcal A_j=I^j/I^{j+1} for every j\in\mathbb Z. The frame p gives the graded conormal frame u=y/p of \mathcal A_1; no ambient lift of u is chosen. Under (A.3), u^{-a} is an absolute dualizing frame \Omega of \omega_E. A change of base root frame has p'=\lambda p, where \lambda is a nowhere-vanishing holomorphic function of the base coordinate. Under this change,

At branch parameters the factors comparing an extending frame of R with these root frames are full-disk holomorphic units, bounded with their inverses on a smaller disk.

A.3 The Hodge test, including classes supported at a point

The moving period equation alone cannot detect a class supported over one parameter. The following test supplies both the growth estimate needed on the punctured curve and detection at the omitted points.

The annular form of the same pairing will produce the differential equation. On a good annulus S in a simply connected good disk, compact-support Leray gives

with kernel H_c^2(S,R^0p_*\mathbb C). Its coherent image factors through H_c^2(S,p_*\mathcal O_Y)=0: the annulus is a Stein curve, and this coherent compact-support group vanishes above degree one. Compatibility of the Leray pairings with trace therefore kills the kernel. The lift of a circle-dual class tensored with a flat \alpha consequently tests s by \oint\langle s/dt,\alpha_{\mathcal O}\rangle\,dt, up to one fixed nonzero normalization. The test is independent of its lift.

A.4 The period equation and all-order lifting

Let g:E\to U be any one of the local full covers above. Push I^j/I^{j+k} forward by the underlying continuous map, as sheaves of complex vector spaces. This convention does not posit a holomorphic map from a thickening to U. We prove simultaneously for every j\in\mathbb Z, every k\ge1, every parameter point and every permitted root choice that

At each fixed order neighborhoods may shrink.

Assume the earlier orders. The current connecting map factors uniquely through the leading layer as \delta_k^j:g_*\mathcal A_j\to R^1g_*\mathcal A_{j+k}: the kernel of the leading-layer map lifts by the preceding induction step. Shorter truncation images have finite filtrations by coherent g_*\mathcal A_i. Acyclicity on small disks and annuli gives leading-section lifts on those entire opens. The overlap difference of products has double-error term of order 2k\ge k+1; hence the collection \delta_k=(\delta_k^j)_{j\in\mathbb Z} is one \mathbb C-linear graded derivation. This remains true in negative degrees because I is invertible. Define e,b\in R^1g_*\mathcal O_E on the full disk by

On the good locus the derivation rule, evaluated in each locally free fiber, gives \delta_k(h)=h'e u^k for holomorphic h. Consequently

Every value of this last obstruction annihilates the constant-sheaf tests, on good annuli and on disks through exceptional points. Indeed its exact sequence is

The first map is dual to reduction of structure sheaves. An obstruction class maps to zero in the middle term; absolute coherent trace is functorial for this closed embedding [29, §§3–5]. Since \mathbb C_E\to\mathcal O_{(k+1)E}\to\mathcal O_E lifts the constant test, the pairing is zero. Only the reduced algebraic family is subjected to Hodge theory; the nilpotent thickening enters through this analytic duality.

Define functional-valued periods \beta_e(\alpha)=\langle e\Omega/dt,\alpha_{\mathcal O}\rangle and \beta_b(\alpha)=\langle b\Omega/dt,\alpha_{\mathcal O}\rangle. The annular trace of (A.12) is

Functions h may have poles at the omitted center because the required lifts exist on the annulus. Integration by parts and the test h=(t-c)^{-1} give the decisive equation

Write \dot\lambda=d\lambda/dt for a base-frame change as in (A.4). The derivation gives

Coordinate Jacobians cancel between \delta_k(t) and \Omega/dt. The actual root-germ cocycles therefore glue the obstruction to

Both periods lie in F^0, so (A.13) says this map is killed by the Higgs field.

Before current-order vanishing, b\Omega is already a holomorphic section of R^1g_*\omega_E on each original full disk. Comparison with its algebraic family, valid also for arbitrary holomorphic frames, gives (A.6) for \beta_b. Radial integration of (A.13) gives

Finite substitutions and the full-disk unit factors preserve this bound in a frame of R^m extending over each puncture. At a branch point, compare on the good punctured overlap, where t=t(w) is unramified. Write \Omega_t for the absolute residue form in the original family and \Omega_w for its transported class there. Then \Omega_w/dw=(dt/dw)\Omega_t/dt, whereas the derivation coefficient satisfies e_w=(dt/dw)^{-1}e_t. The factors cancel. Thus the bound from the original full disk, after finite substitution and the full-disk unit comparison of source frames, introduces no extra pole.

We now use the positivity of the line R^m on the compact curve. More generally, a Higgs-killed map from a positive line to F^0 of a graded-polarizable rational mixed variation with F^1=0 and quasi-unipotent monodromy is zero when its coefficients have the logarithmic-power bound in frames of that line extending over every puncture. To prove this assertion, suppose the map is nonzero and choose the least w for which the image is generically in W_w; holomorphicity makes this containment global. Projection to \operatorname{gr}_w^W is nonzero. Flatness of W_w preserves the derivative condition, and a flat basis adapted to W preserves the coefficient bound. Equip the chosen pure highest Hodge bundle with its Hodge metric H, and denote its Chern connection by D. For a local image section s, let \pi_s be orthogonal projection onto the line spanned by s, away from its zeros. Highest-step curvature gives

[14, Theorem (5.2), Lemmas (4.9),(4.13)]. The incoming Higgs term is absent and the outgoing term vanishes; no flatness of the image line is asserted. A constant Tate twist handles negative Hodge indices. Schmid’s estimate after killing the finite monodromy part bounds flat-vector Hodge norms by powers of logarithms [35, Theorem (6.6')]. Thus, relative to a positive-curvature metric on the source line, the logarithmic norm ratio v has \sqrt{-1}\partial\bar\partial v\ge\omega_R>0 and upper growth O(\log(1+\log(1/|t|))). Subtract a local potential for \omega_R and then \epsilon\log(1/|t|). The maximum principle on shrinking annuli, followed by \epsilon\downarrow0, gives a subharmonic extension preserving this inequality. Interior zeros contribute nonnegative point masses. Integration over the compact curve contradicts \deg R^m>0.

Perfect relative Serre duality now gives e=0 on the good locus; (A.13) and m>0 give b=0. There g_*\mathcal O_E is locally a product of copies of \mathcal O_U: the primitive idempotents of R^0g_*\mathbb C split the proper reduced fibers into connected pieces, and base change identifies each piece’s functions with \mathcal O_U. A derivation kills idempotents and, by e=0, curve functions. It therefore kills every leading section in every degree. At an exceptional parameter, each value of \delta_k^{-m} is now point-supported and annihilates the constant tests. The Hodge test above makes it zero. Applying this to u^{-m} first gives b=0; applying it to f_0u^{-m} for an arbitrary f_0\in g_*\mathcal O_E gives \delta_k(f_0)=0. This includes functions on vertical components, not only pullbacks from U. The Laurent-power rule kills every degree and completes the simultaneous induction (A.11).

A.5 Finite descent and the quadratic count

All conormal lifting obstructions have vanished. We descend only finite collections of layers and count their global sections. This avoids any assertion about convergence of a formal splitting.

For every signed integer j define

On the full cover with deck group H, (\pi_*I^j)^H=J_j: the valuation condition is (r/d_i)\operatorname{ord}_{B_i}(h)\ge j, equivalently \operatorname{ord}_{B_i}(h)\ge\lceil jd_i/r\rceil. Normality checks these divisorial sheaves in codimension one. Since 0<d_i/r\le1, consecutive ceiling differences are zero or one; hence \mathcal B_j is a coherent sheaf on the reduced B. Exactness of finite pushforward and of finite-group invariants, and averaging of the lifts in (A.11), give finite filtrations of f_*(J_l/J_m) with full coherent quotients F_j, for every l<m. The total filtered sheaf is only required to be a sheaf of complex vector spaces.

Cartier periodicity and (A.2) give J_{j-r}=J_j\otimes\mathcal O_V(G) and F_{j-r}=F_j(1). For a coherent Q on \mathbb P^1, a point outside its torsion support gives 0\to Q(b-1)\to Q(b)\to\mathbb C_x^{\operatorname{rk}Q}\to0; thus \chi(Q(b))=\chi(Q)+b\operatorname{rk}Q for every integer b. Finite-filtered cohomology is additive, vanishes above degree one, and satisfies h^0\ge\chi. Consequently

Here \mathcal B_0=\mathcal O_B and the surjectivity of f gives \operatorname{rk}F_0\ge1. Passing through 0\to\mathcal O_V\to\mathcal O_V(nG)\to J_{-nr}/J_0\to0 loses at most the fixed number h^1(V,\mathcal O_V). In particular, the coefficient of n^2 in this lower bound is exactly \tfrac12\sum_{\ell=0}^{r-1}\operatorname{rk}F_\ell>0; all remaining terms are linear or constant. Thus h^0(V,\mathcal O_V(nG)) has positive quadratic growth, contradicting G\sim qA and \kappa(V,A)=0 in (A.1). Every count uses only finitely many infinitesimal orders; no convergent formal neighborhood or one neighborhood valid at all orders is needed. This proves Theorem A.1.