Log abundance in characteristic zero

Proof. Put F=u^*(K_X+\Delta)-v^*(K_{X'}+\Delta'). Discrepancy improvement on divisors of X gives u_*F\geq 0, while -F is u-nef because K_{X'}+\Delta' is nef. The negativity lemma therefore gives F\geq 0. At a prime of W that is not v-exceptional, the coefficient of F is zero unless that prime is extracted on X'. In the latter case it equals the negative of its log discrepancy over (X,\Delta). It is therefore nonpositive, and hence zero. Thus F is v-exceptional. If the source adjoint is nef, then F is also v-nef, so the negativity lemma gives F=0.

The equality of pullbacks transfers semiampleness of a rational adjoint. Indeed a proper birational morphism onto a normal variety has direct image of its structure sheaf equal to that structure sheaf. Projection formula therefore identifies the global sections of each Cartier multiple with the sections of its pullback. If every such pulled-back section vanished over a point downstairs, it could not generate upstairs over that point. Generation upstairs thus implies generation downstairs. Finally, for a klt source the coefficient at an extracted prime would be strictly negative, which is impossible. The remaining discrepancy inequalities show that the model is klt.□

Proof. The ambient variety V is klt. The reduced floor of a \Q-factorial dlt pair is S_2, and it has ordinary double crossings in codimension one; its irreducible components are normal. For the S_2 assertion one can work locally with the cyclic class cover of the \Q-Cartier divisor D. This cover is quasi-étale and klt, hence Cohen–Macaulay. The eigensheaf \OO_V(-D), being a direct summand of its finite direct image, is Cohen–Macaulay. The divisor sequence then shows that D is Cohen–Macaulay.

Divisorial adjunction on the normalization \coprod D_i\to D gives lc pairs (D_i,\operatorname{Diff}_{D_i}(D-D_i)). The different includes the conductor with coefficient one. Removing that conductor contribution defines an effective boundary on D; together with D it is an slc pair whose adjoint line bundle is M|_D. The divisible residue identifications hold in codimension one, with even powers eliminating residue signs at double crossings, and extend by S_2.

Each normalized adjoint is nef and semiample by Assumption 2.1 and Lemma 2.2. Semiampleness descends from the normalization by the slc gluing theorem [26, Theorem 1.5, equivalently Theorem 4.3]. This gives the required statement on the whole reduced scheme, not merely on its individual components.□

Proof. Both \Delta-\epsilon P and \Delta-\tfrac12P are effective klt boundaries. If R is a ray negative for L-\epsilon P, nefness of L shows that R is also negative for L-\tfrac12P. Choose a rational curve C spanning this ray and satisfying the length bound

Writing a=L\cdot C\geq 0 and p=P\cdot C, we have

Our choice of \epsilon gives a<1/m. Since ma is a nonnegative integer, a=0.

For the driving klt contraction c:Z\to B, the line bundle \OO_Z(mL) is numerically trivial over B. The line-bundle clause of the contraction theorem gives its descent to a line bundle \mathcal A_B on B, with no change of m [22, Theorem 3.74(3)]. One may also see the unchanged index directly from relative basepoint freeness: two consecutive sufficiently large powers descend, so their quotient descends. The line bundle \mathcal A_B is nef because its pullback \OO_Z(mL) is nef. For a flip c^+:Z^+\to B, its pullback (c^+)^*\mathcal A_B is the line bundle of the transformed mL. This proves separately that the transformed L is nef and that mL remains Cartier with the same m.

The driving boundary remains klt throughout its LMMP. The transformed \Delta-\tfrac12P remains effective and is smaller than the driving boundary, so it too is klt. The same estimate and the same integer m apply inductively at every subsequent step.□

Proof. Hashizume’s reduction gives nonvanishing and log minimal models for projective lc pairs with real boundary in dimensions at most n [36, Theorem 1.4]. Thus we may pass to a \Q-factorial dlt log minimal model. Fix the support of its boundary, impose the rational affine constraints that its coefficient-one components remain equal to one, and take a sufficiently small rational polytope around the given boundary within those constraints. On a fixed log resolution witnessing dlt, the strict discrepancy inequalities remain strict in this neighborhood; thus its boundaries remain dlt. The rational-polytope theorem for nef adjoints [4, Remark 3.1 and Proposition 3.2(3)], applied to all extremal rays and intersected with this neighborhood, expresses the given real boundary in a finite rational simplex of dlt boundaries with nef adjoints. It is enough to prove semiampleness for these rational adjoints. Their positive real combinations are semiample, using the product of the associated contractions. For a rational adjoint, real semiampleness can also be rationalized: the finite linear equations expressing its divisor and principal-divisor coefficients have rational data, so a real solution with positive coefficients has a rational solution with the same positivity.

We therefore work with a rational dlt pair (Z,\Delta) and L=K_Z+\Delta nef. Real nonvanishing gives rational nonvanishing in this case: retain the finitely many divisors and principal divisors in an effective real representative and solve the resulting rational linear system with nonnegative rational coefficients. If \kappa(Z,L)\geq 1, lower-dimensional good models give a good minimal model by [28, Lemma 3.5]. Lemma 2.2 transfers semiampleness back to Z. It remains to treat

The non-pseudo-effective perturbation. Put P=\Delta in the klt case and P=\lfloor\Delta\rfloor in the other case. Suppose L-\epsilon P is not pseudo-effective for every sufficiently small \epsilon>0. Choose rational \epsilon as in Lemma 2.4, and run the klt LMMP with scaling to a Mori fiber space

Existence and termination in the non-pseudo-effective case are the established klt results of [6]. The program is crepant for L. The line-bundle descent in Lemma 2.4, applied also to the final contraction, gives a nef rational divisor A on T with

The base has dimension less than n.

If (Z,\Delta) is klt, the transformed original pair (Z',\Delta') is klt as well. Ambro’s descent theorem [1, Theorem 0.2] gives an effective rational boundary \Delta_T such that (T,\Delta_T) is klt and

All its hypotheses hold: the total pair is globally klt and effective, f is a projective contraction, and its adjoint is rationally linearly pulled back. In particular, this use requires no conjecture about semiampleness of a moduli divisor. Assumption 2.1 and Lemma 2.2 make A semiample.

Otherwise P' is effective and relatively ample, since K_{Z'}+\Delta'-\epsilon P' is relatively antiample. Some component of the floor therefore dominates T. Take a crepant dlt blowup b:(\widetilde Z,\widetilde\Delta) \to(Z',\Delta'), and choose the strict transform S of such a component. The restriction f_S=(f\circ b)|_S:S\to T is surjective. Crepancy and adjunction give an lc pair on S whose nef adjoint is rationally linearly equivalent to f_S^*A. It is semiample by the lower-dimensional hypothesis. Semiampleness of a rational line bundle descends along a proper surjection: use the equality of sections for its connected-fiber Stein factor, and norms for the remaining finite morphism. For the latter assertion, choose a section of a basepoint-free multiple nonzero simultaneously at all points of the chosen finite fiber. Such a section exists because each vanishing condition is a proper linear subspace, and finitely many proper linear subspaces do not cover a complex vector space. Its norm is nonzero at the point downstairs. Hence A is semiample also in this case. Crepancy transfers the conclusion back to L.

The klt pseudo-effective perturbation. We next settle the klt case when L-\epsilon\Delta is pseudo-effective for some small rational \epsilon>0. Nonvanishing gives

Take a resolution p\colon W\to Z with reduced SNC divisor D consisting of all exceptionals and the strict support of H_0+\Delta. We have

Here every component of D has positive coefficient in

Indeed, at strict boundary components the difference between coefficient one and a klt boundary coefficient is positive, and at exceptional components the coefficient contributed by the adjoint comparison is the positive log discrepancy. The pushforward p_*G_W is bounded coefficientwise by a fixed multiple of G_0. Section spaces inject under birational pushforward, so

Take a \Q-factorial dlt log minimal model (V,D_V) of (W,D), and fix a common resolution

Its boundary is reduced. On U, the comparison divisor in Lemma 2.2 is exceptional over V. Pushing q^*G_W forward by v therefore gives

For completeness, positivity at a component extracted on V also holds: its log discrepancy over (W,D) is zero, its center lies in D, and the pullback of the full-support effective G_W has positive order there. There is no comparison-divisor coefficient at a prime retained on V. Effectivity and exceptionality in the comparison preserve the adjoint section ring, so the nef adjoint on V still has Iitaka dimension zero.

The small-perturbation hypothesis of Theorem 3.1 is now explicit. If D_V\neq 0, choose rational c with 0<c<\min_i b_i. Then

If D_V=0, pseudo-effectivity is immediate. Proposition 2.3 supplies the required semiampleness on D_V. Theorem 3.1 thus makes K_V+D_V abundant. Its Iitaka dimension is zero, so its numerical dimension is zero as well. For an ample divisor H_V on V, we obtain

Effectivity forces G_V=0, and its full support then gives D_V=0.

Recall that G_W=p^*G_0+R with R\ge0. On the same common resolution,

Thus q^*p^*G_0 is effective and v-exceptional. It is nef because it is rationally linearly equivalent to (p\circ q)^*L. The negativity lemma for v forces it to vanish. Therefore G_0=0 and L\sim_{\Q}0. Together with the preceding cases, this proves the klt good-model assertion in dimension n: for a non-nef pseudo-effective klt adjoint, first take its klt log minimal model and apply what was just proved.

The remaining non-klt case. Finally suppose the original pair is not klt and a small floor perturbation L-\delta P is pseudo-effective. Choose rational \epsilon>0 with 2\epsilon\le\delta and small enough to keep the perturbed boundaries effective and klt. Convexity of the pseudo-effective cone, applied between L and L-\delta P, then makes both

pseudo-effective klt adjoints. Their Iitaka dimensions are zero: nonvanishing gives the lower bound, and adding the effective perturbation bounds each by \kappa(Z,L)=0. The preceding klt case gives good models for both of these perturbed adjoints. Consequently their Nakayama numerical dimensions are zero. Since

and the last summand has an effective rational representative, monotonicity and homogeneity of \kappa_\sigma give

The nef divisor L is therefore numerically trivial. Nonvanishing is already available in this finishing step: write L\sim_{\Q}E\geq 0. For an ample divisor H, the equality E\cdot H^{n-1}=0 forces E=0. Thus L\sim_{\Q}0, without any additional nonvanishing or abundance premise.

This completes the remaining zero-Iitaka-dimension case, so every rational nef dlt adjoint considered above is semiample. Returning to the finite rational simplex proves the real-boundary good-model assertion. Lemma 2.2 then descends semiampleness to every original nef rational lc pair.□

Proof. For v=n, the numerical criterion for a nef divisor gives bigness. Assume v<n. Intersect the pseudo-effective class L-cD with the indicated product of nef classes:

Each number D_iL^vH^{n-v-1} is nonnegative, so all of them vanish. When v=0, ampleness then forces D=0; the signed representative consequently gives L\sim_{\Q}0.

For v>0, semiampleness identifies the numerical dimension of the restriction to each component of D with that component’s image dimension under \phi. The vanishing just proved bounds this dimension by r. Moreover,

so the bound is attained by a component with positive coefficient.

To control where that positive part meets the rest of the boundary, consider the symmetric matrix

Distinct effective \Q-Cartier divisors have effective intersection cycles, so the off-diagonal entries are nonnegative. The ambient intersection form has at most one positive direction by the mixed Hodge index theorem: approximate by ample classes and apply the ordinary Hodge index theorem on complete-intersection surfaces. In this form L is isotropic and orthogonal to every D_i, while its pairing with H is strictly positive. Indeed, writing Q also for the ambient bilinear intersection form, we have

whereas Q(L,H)=L^{r+1}H^{n-r-1}>0 because r+1=v. The form on the orthogonal space to L is therefore negative semidefinite. In particular this holds on the span of the D_i. Pulling back to a resolution gives the same conclusion, so smoothness of X is unnecessary here.

Write a=(a_i) for the coefficient vector. The signed relation gives the vector equation Qa=0: for every i,

Temporarily write a=a^+-a^- for its decomposition into positive and negative coefficient vectors. We obtain

Negative semidefiniteness forces both sides to vanish. A vector on which a negative semidefinite quadratic form vanishes belongs to its kernel, so Qa^+=0. In a row with a_i\le0, the diagonal term is absent and every summand Q_{ij}a_j^+ is nonnegative. Their sum is zero; hence each intersection with a positive component has zero L^r-degree against H^{n-r-2}. Semiampleness on D then makes its image dimension less than r. For r=0, any nonzero effective intersection would have positive ample degree, so the intersection is empty.□

Proof. We construct the divisor and its adjunction data first, then place its Stein image in a smooth projective bundle. The final step verifies that the infinitesimal neighborhoods have the scheme charts required for formal lifting.

Roots and a fixed adjunction isomorphism. Let \rho:\mathcal R\to X be the normalized simultaneous root stack of the Cartier divisors mG_+, mG_-, and mD_0, taking an \ell-th root of each divisor together with its section; the construction allows empty divisors. Denote the tautological root divisors by S_+,S_-,S_0 on \mathcal R. Their reducedness can be checked at a generic prime of downstairs multiplicity b. The chosen divisibility conditions give b\mid\ell. A normalized chart of y^\ell=x^b has ramification index \ell/b, and y has order one. Since m divides \ell, this also applies to D_0. The divisors are Cartier, so normality and generic reducedness give their reducedness everywhere.

Put B=S_++S_-+S_0 on \mathcal R. With this full reduced boundary, log ramification gives a log canonical pair and the relation

The ambient charts are klt. They are klt off the boundary, and, on a log resolution, decreasing every boundary coefficient slightly removes all zero-discrepancy places. The difference between the two sides is a torsion integral Weil class, represented by

Choose a periodicity isomorphism \mathcal F^{[q]}\simeq\OO_{\mathcal R}, where square brackets denote reflexive powers. It defines multiplication in the finite algebra

Let \tau:\mathcal R^\dagger\to\mathcal R be the normalization of \Spec_{\mathcal R}\mathcal A. In codimension one this is the ordinary cover of a torsion line bundle and is etale. The cover preserves log canonicity, klt ambient singularities, and the reduced Cartier boundary. From now on S_+,S_-,S_0 denote their pullbacks to \mathcal R^\dagger.

Tautological evaluation on this cover trivializes the pullback of \mathcal F in codimension one and hence fixes the adjoint isomorphism. Because this evaluation is a sheaf morphism on the cover stack itself, the adjoint isomorphism is equivariant on every atlas. This equivariance will be needed for the quotient below.

Separating the positive and negative parts. Let p_1:\widetilde{\mathcal R}\to\mathcal R^\dagger be the normalized blowup of the ideal of S_+\cap S_-. The spaces constructed so far are related by

The blown-up ideal is locally generated by two elements, so the ordinary blowup embeds in a relative projective line. Its fibers have dimension at most one, as do the fibers after finite normalization. Hence each exceptional divisor lies over a codimension-two component of the intersection. Downstairs such a component is a dlt stratum, generically SNC. Separate normalized Kummer charts and purity for the index cover at the smooth generic locus give the same description upstairs.

The tautological exceptional divisor E is consequently reduced Cartier, and the divisors

are reduced Cartier and disjoint on \widetilde{\mathcal R}. Since no codimension-two two-branch stratum is contained in S_0, its pullback p_1^*S_0 is reduced Cartier and has no exceptional component. The total boundary

is crepant and log canonical. To check the ambient singularities as well, denote this boundary on a chart V' by B'. The crepant equality gives log canonicity for every valuation. Outside B', the blowup is an isomorphism to the old klt boundary complement. A divisor F with a(F;V',B')=0 therefore has center in \Supp B'. Since B' is effective Cartier, \ord_F(B')>0, whence

Positive pair discrepancies also remain positive after the boundary is dropped. Thus V' is klt even at its higher-codimension singular loci. On \widetilde{\mathcal R}, put S=S'_+ and T=S'_-+E+p_1^*S_0. Near S, the section of the disjoint divisor S'_- is a unit, and the fixed linear equivalence reads

Klt singularities make the ambient charts Cohen–Macaulay. All the boundary divisors in this display are Cartier, so the isomorphism also makes these charts Gorenstein.

We also need finiteness of S\to S_+ on \mathcal R^\dagger. Locally write S_+=(u=0) and S_-=(v=0). Before normalization, the strict transform of S_+ lies in the blowup chart with ratio u/v, where its equation is u/v=0. Thus each fiber of this strict transform over S_+ has at most one point. Properness and finite normalization prove finiteness of S\to S_+. A codimension-one point of S\cap T therefore lies over a codimension-two intersection downstairs. The generic SNC description shows that T|_S is generically reduced. Being Cartier on the Cohen–Macaulay scheme S, it is reduced everywhere.

Retaining the required root line. The construction so far has supplied reduced Cartier divisors and fixed adjunction data. We now remove the root characters that are unnecessary, while preserving the line attached to S. On \widetilde{\mathcal R} the line \OO(S-S'_-) has \ell-th power equal to the pullback of N_0 and hence defines a morphism to the gerbe of \ell-th roots of N_0 on X. Denote this gerbe by \mathcal G, and let \kappa:\widetilde{\mathcal R}\to\mathcal X be the relative coarse-space morphism over \mathcal G. On a trivializing chart, we quotient by the kernel of the finite-group character on the indicated root line. The quotients are ordinary quasi-projective schemes: the covers used above are finite, the blowup is projective, and finite quotients preserve quasi-projectivity. These chartwise quotients patch.

Before taking the quotient, the section of the disjoint divisor S'_- is a unit near S. There the retained line \OO(S-S'_-) identifies with \OO(S), so the line of S and its section both descend through the kernel quotient. Continue to write S,T for their reduced images under \kappa; they now lie on \mathcal X. In particular S remains Cartier. The finite-group norm of a local equation for T upstairs makes its image set-theoretically principal; Cartierness of the reduced image T is not needed. Divisorial ramification occurs only on the boundary. Log ramification thus preserves log canonicity and identifies the descended adjoint isomorphism with (\ast_{\mathrm{adj}}). The ambient chart and S are Cohen–Macaulay because their structure sheaves are invariant direct summands of the finite CM covers.

More explicitly, the kernel acts trivially on the retained root line and hence on \OO(aS). Equivariance of the fixed isomorphism gives the same kernel character on \omega(S+T). Taking invariants descends the isomorphism, and log ramification identifies the invariant log dualizing sheaf. Off T, this isomorphism makes the canonical sheaf invertible, giving the asserted Gorenstein property.

Duality on the boundary. The reduced supports S and J are unions of log canonical centers, since intersections of lc centers are again unions of lc centers. Both are therefore Du Bois by [43, Theorems 1.4 and 1.7]. To identify the ideal and its dual, work on a quotient chart and denote the finite cover above by f:S^{\mathrm{up}}\to S. Reducedness of T^{\mathrm{up}}|_{S^{\mathrm{up}}} gives

An invariant function vanishes on the reduced quotient image exactly when its pullback vanishes on this reduced preimage. The displayed invariant sheaf is maximal Cohen–Macaulay: the upstairs sheaf is invertible on a CM scheme, finite pushforward preserves its depth over the quotient, and in characteristic zero invariants form a direct summand.

Apply finite-map duality with trace and then take invariants:

The last isomorphism comes from Cartier adjunction upstairs and descent of the line of S. Finite-map duality applies in the presence of ramification; it requires no invertibility of \omega_S itself. Off J, the identification is the ordinary residue fixed by the ambient isomorphism. Normalized trace preserves this residue convention, so the identifications agree on overlaps.

The target of the lifting argument. On S, the root gerbe of N_0 is D\times_P\mathcal P. Finiteness of the strict divisor, proved above, and removal of the relative inertia kernel give a finite representable map S\to D\times_P\mathcal P. Its image covers the positive boundary, and the root line is \OO_S(S)=C. Lemma 3.2 therefore gives image dimension r for S in P. To check the smaller image of J, recall that before the quotient S'_- is disjoint from S, the exceptional divisor E maps into \Supp G_+\cap\Supp G_-, and p_1^*S_0 maps into D_0. Taking the quotient changes none of these images in X. The image of J in D is consequently contained in

The same lemma gives image dimension less than r for this locus in P. We have thus obtained a representable projective morphism h:S\to\mathcal P; it remains to embed its Stein image in the required smooth bundle.

The Stein algebra defines a finite stack \mathcal T=\Spec_{\mathcal P}h_*\OO_S over \mathcal P. This coherent algebra is generated as a module by vector bundles on \mathcal P. Indeed, decompose it by the finitely many inertia characters, twist each summand by a power of C so that it descends to P, and use Serre generation there. The resulting vector-bundle surjection embeds \mathcal T in a vector bundle over \mathcal P. Since \mathcal T is proper over the base, it remains closed in the projective completion. This completion is \mathcal Y and gives the map g:S\to\mathcal Y from the reduced divisor. The Stein construction yields g_*\OO_S=\OO_{\mathcal T}, and the line identities follow from those already established.

Finally, start with the etale chart S_U\to S obtained from g. Invariance of the etale site under nilpotent thickening extends this chart uniquely to qS_U\to qS, compatibly as q varies. This construction does not use a map qS\to\mathcal Y; lifting the map from S is the task of the next section. The reduction S_U is a scheme; its extension qS_U is an algebraic space with trivial inertia. The chosen chart is separated, the map to the root gerbe is finite, and that gerbe has finite diagonal, so qS_U is separated over X. The finite representable map S\to D\times_P\mathcal P and the etale map S_U\to S show that the finite-type geometric fibers of S_U\to X are zero-dimensional. Nilpotent thickening changes neither geometric points nor fiber dimensions. The morphism qS_U\to X is therefore quasi-finite as well. Zariski’s main theorem embeds qS_U in a scheme finite over X, making it a quasi-projective scheme as required.□

Proof. The proof must identify not just a coherent sheaf but its actual inclusion into the differential module: the graph calculation will differentiate classes in that inclusion. We first compute the sheaf by duality. Du Bois comparison applied to the difference triangle for the constant objects of S_U and J_U identifies its degree-zero graded de Rham complex with \mathcal I_{J_U/S_U}. Coherent duality and the maximal-Cohen–Macaulay property give

Here [N] cancels the dimension shift of the dualizing complex. In the smooth case, our convention gives \mathbf D(\Q^H[N])=\Q^H[N](N); its right differential module has \omega as its lowest piece, at index zero.

To pass to perverse cohomology, work locally on a fixed chart and let q be the smallest filtration index occurring in any perverse cohomology object of A^\bullet_U. Boundedness of the complex and goodness of the filtrations provide such a lower bound. At index q, each of these objects has a graded Spencer complex consisting only of its degree-zero term. The spectral sequence for perverse truncation has a single nonzero row at that index, and therefore gives

If q<0, this contradicts Equation (3). It follows that all these objects have F_{<0}=0. The same argument at index zero yields F_0A_{0,U}=\mathcal B|_{S_U}, while the lowest pieces of the other perverse cohomology objects vanish. This passage does not require the shifted constant complex to be perverse.

We next identify the underlying unfiltered module. Ambient duality in V identifies the dual of the reduced constant complex with

The degree-zero module is the first nonzero support cohomology, \mathcal H^c_{S_U}(\omega_V). The support of J_U is locally set-theoretically principal in S_U by the geometric construction. Lift a local defining function to V and invert it. This exact meromorphic localization realizes j_* on the underlying module and gives the description of A_{0,U} in the statement.

The two descriptions must agree as inclusions, with the fixed residue normalization. On the smooth locus of S_U\setminus J_U, filtered duality and smooth graph pushforward give the ordinary dualizing sheaf and its residue inclusion. This is precisely Ext-to-support with the normalization specified by ambient adjunction. Naturality in smooth etale coordinates gives the same scalar on each chart. The agreement extends throughout S_U\setminus J_U, because the first support-cohomology module has no sections supported on a smaller-dimensional subset. One can see this from the Cousin resolution of the smooth dualizing sheaf: its first term supported on S_U is a sum of injective modules at the codimension-c generic points. Localization extends the agreement across J_U. We have therefore identified the actual lowest-piece inclusion, including its residue convention.

We have now identified F_0A_{0,U} and its inclusion before direct image. To identify F_0M_U, compute g_+ by a graph embedding followed by a smooth projection. At filtration zero, only the top term of the relative Spencer complex survives: it is the direct image of \mathcal B|_{S_U} on the graph. Projective strictness yields

and identifies this sheaf as a coherent subsheaf of the unfiltered degree-one direct image. All constructions in this identification are canonical under etale changes of charts.□

Proof. The line C lives on the root gerbe, so we first push the complex to \mathcal P. We will then pull back to projective space, where C becomes \OO(1) and ordinary Kodaira–Saito vanishing applies. Since p is finite on the support of M, its direct image is concentrated in perverse degree zero; call the resulting system on \mathcal P M'. On ordinary scheme charts, projective strictness gives

For the desired global hypercohomology vanishing, this identity must hold for the descended complexes. We verify that compatibility before passing to a finite cover.

For right differential modules, use the transfer bimodule

with its order filtration. Derived tensor over \mathcal D_{\mathcal Y}, followed by derived direct image, forms the filtered direct image; equivalently, these operations can be carried out on Rees modules. De Rham on the base is a further derived tensor with \OO_{\mathcal P}, filtered from degree zero. The filtered Spencer resolution, associativity of derived tensor, and projection formula identify this with direct image of the de Rham complex upstairs. Indeed, tensoring the transfer module over the base differential operators with the base structure sheaf gives \OO_{\mathcal Y}.

These sheaf constructions are canonical on the etale site, and the Spencer resolutions are locally free over differential operators. They commute with chart changes already before taking associated gradeds. The ordinary projective direct-image theorem on the base charts supplies strictness and concentration in perverse degree zero. Passing to the etale site does not change coherent cohomology on those charts. This proves Equation (4) globally, as an identity computing coherent hypercohomology. The same verification will apply to the finite representable map below.

To pass from the gerbe to a scheme, use the representable finite surjective morphism

defined by the power map [z_0:\cdots:z_s]\mapsto[z_0^\ell:\cdots:z_s^\ell] and the root \OO_{\PP^s_z}(1); thus v^*C=\OO_{\PP^s_z}(1). Trivial roots on the standard affine opens of \PP^s give scheme etale charts \check U_i\to\mathcal P. The restriction of v to z_i\ne0 factors through \check U_i. On each such chart take the derived Hodge-module inverse image of M' and then its degree-zero perverse cohomology. Thus the object being constructed chartwise is

The chosen root and functoriality identify these objects on overlaps, producing a system on the Zariski opens of \PP^s_z.

We check that H is a global graded-polarizable mixed Hodge module, so that ordinary vanishing applies to it. The perverse, filtered, and weight data glue, as do the strict-support decompositions of the pure weight gradeds, by uniqueness. On a smooth connected dense stratum of an irreducible support, a polarization on a nonempty Zariski open extends to a flat pairing: the fundamental group of that open surjects onto the fundamental group of the stratum. Hodge compatibility extends by continuity, and the extended flat pairing remains nondegenerate and positive. The local Hodge modules already give boundary quasi-unipotence. Saito’s extension theorem for polarizable variations [56, Section 3.b] and uniqueness of strict-support extension identify the global pure object with the glued object. Projectivity of \PP^s_z leaves no additional boundary at infinity. The weight extensions give the required mixed object.

Vanishing for H will imply vanishing for M' once we realize M' as a retract of v_+H. On a chart one must use the whole base-changed finite cover. Over \check U_i, this is a disjoint union of copies of the corresponding affine coordinate chart of \PP^s_z. The unit on unshifted constants and its dual trace, formed using smooth duality in equal dimensions, compose to multiplication by the degree. This can first be checked on the finite etale locus, where the trace sums over the sheets. On each connected smooth base chart, the shifted constant Hodge module is the rank-one intersection complex, with endomorphism ring \Q, and its endomorphisms are determined on a dense open. The composition therefore equals the degree globally, with no contribution supported on the branch locus. This argument permits ramification.

Tensor the unit and trace with M', apply proper projection formula, and take perverse cohomology in degree zero. Finite direct image is perverse exact, so it commutes with this degree-zero cohomology. We obtain maps

whose composite is multiplication by the degree of the whole base-changed cover. Dividing the second map by that degree gives the retraction. Each map is canonical under etale base change; the retraction consequently descends on filtered differential modules as well.

Applying the finite version of Equation (4) and projection formula now exhibits the hypercohomology in the statement as a direct summand of

Ordinary negative-ample Kodaira–Saito vanishing makes this group zero for j<0. For a mixed object, apply the vanishing to the pure weight gradeds and then to the weight-filtration exact sequences, whose Hodge filtrations are strict.□

Proof of Proposition 4.1. We induct on k, simultaneously for all powers I^j. The connecting homomorphisms will factor through the first graded layers and together form a derivation. We must show that this derivation vanishes both on functions from U and on a local generator of I/I^2. The graph calculation relates these two errors in the differential module M_U. Its order-one symbol, together with negative-twist vanishing, first kills the function error. The original relation then kills the conormal error.

Define

The first graded layer is

For adjacent lengths, the exact sequence has kernel g_*\OO_{S_U}\otimes C^{-j-k}, and its connecting homomorphism takes values in

The obstruction derivation. Assume inductively that all shorter transitions are surjective. A section of E_{j,k} with zero length-one term comes from E_{j+1,k-1} when k>1. Induction lifts it locally from E_{j+1,k}, so its connecting class vanishes. Induction also makes E_{j,k}\to E_{j,1} surjective. The obstruction thus factors through the length-one layer; for k=1 this factorization is immediate. More precisely, these factorizations give additive maps

Together they define D_k on the graded algebra \bigoplus_{j\geq0}\OO_{\mathcal T_U}\otimes C^{-j}, with graded shift k; no \OO_U-linearity is asserted. Its values are computed by choosing local lifts and taking Cech differences. For a product, this difference is the sum of the two first-order differences: the product of two errors vanishes in the layer under consideration. Thus D_k is a \C-derivation, with values in the graded module whose degree-j term is R^1g_*\OO_{S_U}\otimes C^{-j}.

Restrict its degree-zero part along \OO_U\to\OO_{\mathcal T_U}. This derivation factors through \Omega_U^1, defining a section of T_U\otimes R^1g_*\OO_{S_U}\otimes C^{-k}. As the chart varies, these sections descend to

The equality follows from Equation (2), Lemma 4.2, and projection formula. For the descent assertion, function lifts pull back along the unique etale thickenings. Ordinary flat base change on the reductions pulls back their Cech classes in the coherent obstruction sheaves. Differentials also pull back and generate under an etale map, giving the required compatibility.

Coordinate errors and adjunction. We now compute this derivation in local coordinates. After shrinking U, choose etale coordinates t_1,\ldots,t_d, with d=\dim U, and a frame of C^{-1}. Write y for its pullback, a conormal frame. Induction lifts the coordinates along g modulo I^k and lifts y to I/I^{k+1}. Lift them one step further on an open cover of S_U, obtaining functions h_{i,\lambda} modulo I^{k+1} and generators y_i modulo I^{k+2}. The conormal generator must be lifted one order farther than the coordinates: its next error lies in I^{k+1}/I^{k+2}, whereas a coordinate error lies in I^k/I^{k+1}. By formal etaleness of the coordinate map, each h_i defines a local map from (k+1)S_U to U. Write the overlap errors as

Thus D_k(t_\lambda) is represented by (e_{ij,\lambda}y^k), and D_k(y) by (\eta_{ij}y^{k+1}). The powers of y record the target line bundles; e_{ij,\lambda} and \eta_{ij} are their coefficients on the reduction. Denote the induced frame of \mathcal B|_{S_U} by \sigma=y^{-a}.

Off J_U, the fixed adjunction isomorphism gives

Via this isomorphism, the local equation y_i defines a dualizing generator, denoted b_i=y_i^{-(a+k)}. This notation denotes a frame of the line bundle \OO_{\mathcal X}((a+k)S) restricted to the thickening; it does not invert the nilpotent function y_i in its structure sheaf. Realize the dualizing sheaves in support cohomology. Under the trace inclusion \omega_{S_U}\hookrightarrow\omega_{(k+1)S_U}, we have

The first identity is the binomial expansion of (1+\eta_{ij}y_i^k)^{-(a+k)}: all quadratic terms vanish because 2k\geq k+1. In the last two identities, \sigma is viewed in the dualizing sheaf of the thickening through the trace inclusion. They express Cartier trace and the annihilator of the thickening.

The graph calculation. We next put both errors in the same differential module. Embed (k+1)S_U as a closed subscheme of a smooth open Z\subset\overline Z=\PP^m. The graph of the reduced map g is closed in \overline Z\times U, since it is proper over U. Let \iota:S_U\hookrightarrow\overline Z\times U be this reduced graph embedding, and let \mathcal N_\Gamma be the underlying unfiltered differential module of \iota_+A_{0,U}. This module is distinct from the direct-image module M_U on U. For each local graph lift h_i, Ext-to-support cohomology sends the dualizing section b_i to a section \delta_i(b_i) of \mathcal N_\Gamma. All the lifted graphs have the same reduced support, namely the graph of g, so these sections lie in one support-cohomology module rather than in different modules for the different lifts. These sections need not lie in its lowest filtered piece. We will prove the identity

On the right, each reduced dualizing section \sigma is included in \mathcal N_\Gamma by the lowest-piece map of Lemma 4.2. The right differential-operator action is taken in this graph module.

The calculation also applies when S_U is singular. Put c_0=\dim Z-N. Ext-to-support identifies the dualizing sheaf of the thickening with its annihilator submodule in \mathcal H^{c_0}_{S_U}(\omega_Z). To see this, use the support-cohomology spectral sequence. Support cohomology vanishes below c_0, so in total degree c_0 the Ext group consists of homomorphisms from the structure sheaf of the thickening to the first support-cohomology module. In particular, no local complete-intersection hypothesis on S_U\subset Z is needed.

Lift the h_{i,\lambda} locally to functions on Z. For a dualizing section b of the thickening, its graph inclusion in Z\times U is the generalized fraction

It represents successive support cohomology in the additional coordinate equations, equivalently the Koszul residue for the graph, and can be computed etale-locally near the graph branch. After taking support cohomology from Z, the translated new coordinates form a regular sequence; their support cohomology is consequently concentrated in the degree equal to their number. Either set of translated coordinates gives the same localization of this support-torsion module, because the two translations differ nilpotently on every section.

The change from h_i to h_j thus has a finite Taylor expansion on b_j. Again using 2k\geq k+1, products of two differences from Equation (6) annihilate b_j. The linear terms are determined by

The right action on a top differential form satisfies

The doubled pole therefore contributes the positive sign in Equation (8). Combining these terms with Equation (7) proves the identity. So far the calculation is off J_U. Both its sections and its identities extend meromorphically across J_U in the unfiltered localized module, where arbitrary finite pole orders are allowed.

Vanishing of the obstruction. Push the graph identity along the projection \pi:\overline Z\times U\to U, using the relative Spencer complex. Closed direct image is perverse exact, and \pi_+\iota_+A_{0,U}=g_+A_{0,U}. Thus degree-one holonomic differential-module cohomology of this Spencer direct image is the underlying module of M_U={}^{p}\!H^1g_*A_{0,U}. The cochain (\delta_i(b_i)) belongs to the top, degree-zero term, from which there is no outgoing relative Spencer differential. Its Cech difference is a boundary in total degree one. A sufficiently refined ambient cover around the closed graph computes this boundary; all the sheaves in question are supported there. By Lemma 4.2, the reduced cocycles in Equation (8) represent their R^1g_*\mathcal B classes in F_0M. Right differentiation in the U-coordinates acts on the pushforward complex. Consequently the Cech classes satisfy

Here each bracketed class belongs to R^1g_*\mathcal B=F_0M_U; after differentiation, its image belongs to F_1M_U. Thus every term of the relation lies in F_1M, and the \eta-term lies in F_0M. The equality itself was proved in the unfiltered differential module. The inclusion F_1M\subset M is an inclusion of actual sheaves, so vanishing in M implies vanishing in F_1M. Taking the order-one symbol shows that the global section e from Equation (5) is killed by

after twisting by C^{-(a+k)}. Only the relation is used in this filtered step; no bound on the Hodge filtration of the auxiliary cochain (\delta_i(b_i)) has been imposed.

Because F_{<0}M=0, the entire complex at filtration index one is

in degrees -1 and 0, with differential the symbol map in Equation (9). After the indicated twist, its degree--1 hypercohomology is therefore exactly the space of global sections of the kernel of that map. The twist is negative, since a+k>0. Lemma 4.3 annihilates this space, and hence e=0.

It remains to kill the conormal part of the derivation. With coordinates and a line trivialization fixed locally, e=0 says that every class [e_{ij,\lambda}\sigma] vanishes in F_0M, and thus in M. Its actual differential-operator image vanishes as well, before passing to symbols. Returning to the unfiltered relation gives

Injectivity of the lowest-piece inclusion and a+k>0 then give D_k(y)=0. The degree-zero derivation also vanishes: its restriction to \OO_U is the coordinate error e, and \OO_U\to\OO_{\mathcal T_U} is surjective because \mathcal T_U is a closed subscheme of U. The coefficient algebra and the conormal frame y generate the full graded length-one algebra locally. Thus D_k=0 in every degree, completing the induction and proving surjectivity of all the sheaf transitions.

The kernels are the stated coherent sheaves \OO_{\mathcal T_U}\otimes C^{-j-k}. If U is affine, their first cohomology vanishes, so the exact sequences of sheaves of abelian groups give surjectivity on global sections as well. Successive choices of lifts then give compatible formal lifts of any chosen functions on \mathcal T_U and, after trivializing C, of the transverse conormal frame.□

Proof. We construct a formal deformation of a general boundary fiber, algebraize it over \C[[s]], and show that the resulting subvarieties belong to a family dominating X.

Deforming a boundary fiber. Choose a separated affine etale chart U\to\mathcal Y near a general point of a top-dimensional component of its Stein image \mathcal T_U. After shrinking, C is trivial, that component is smooth of pure dimension r, and S_U is flat over it. Since the image of J in P has dimension less than r, we may also arrange that S_U is disjoint from J_U. Choose a closed point t in this open set. After a further shrinking, regular parameters t_1,\ldots,t_r on \mathcal T_U cut out only t. The fiber

is projective and has pure dimension d. Along F, the pullbacks of the parameters form a regular sequence. The ideal sheaf \mathcal I_{J/S} is maximal Cohen–Macaulay by Proposition 3.3(ii). Since S_U is disjoint from J_U, the etale pullback of this ideal sheaf is \mathcal I_{J_U/S_U}=\OO_{S_U}; hence S_U is Cohen–Macaulay. The quotient F by the regular sequence is Cohen–Macaulay as well. We will use this property to control the dimensions of the components after algebraization.

Write I=\OO_{\mathcal X}(-S). By Proposition 4.1, the transitions between the sheaves g_*(I^j/I^{j+k}) are surjective as the length k increases. Their kernels are the coherent sheaves g_*\OO_{S_U}\otimes C^{-j-k}. Affineness of U also gives surjectivity on global sections. We can therefore lift the parameters compatibly through all thickenings, and lift the chosen conormal frame to a compatible element \widetilde y that formally generates I.

Let Z_q be the closed subscheme of qS_U cut out by the r lifted parameters. Regard it as a scheme over

The construction has the diagram

The middle vertical map is defined by the lifted normal parameter. The lifted base parameters cut out Z_q; they are not provided by a preexisting morphism of the ambient thickening to the base.

The schemes Z_q are compatible under reduction, have closed fiber F, and are flat over R_q. For flatness, the powers of the Cartier generator first identify the successive s-layers of qS_U with \OO_{S_U}. This proves flatness before imposing the parameters. The local flatness criterion then preserves flatness when we quotient by lifts of the regular sequence on the closed fiber.

Algebraization and boundary avoidance. The map Z_q\to X\times\Spec R_q is quasi-finite. It is also proper: its reduction is the map from the projective scheme F, and nilpotent thickenings do not change properness of a finite-type morphism. Thus it is finite. Put R=\C[[s]]. Projective Grothendieck existence algebraizes the compatible finite algebra sheaves on X\times\Spec R_q to a finite algebra on X_R=X\times\Spec R. Full faithfulness algebraizes their multiplication and unit [62, Lemmas 30.24.1 and 30.24.3]. The relative spectrum gives a finite morphism

with exactly the prescribed reductions. In particular, Z is projective over R. Formal flatness gives flatness along the closed fiber; on the generic fiber over \C((s)) it is automatic. Hence Z is flat over R.

The images of the pole and coefficient-zero boundary parts miss F. Their inverse images in Z are closed and proper over R, and must therefore be empty: any nonempty closed subset of a proper R-scheme specializes to the closed fiber. Recall also the exceptional divisor E of the normalized blowup separating S_+ and S_- in Proposition 3.3. Its image in X lies in \Supp G_+\cap\Supp G_-, so it is already excluded by avoidance of the pole part.

On the formal neighborhood from which the Z_q were cut out, the root construction therefore identifies the divisor of the pulled-back section s_0 with \ell S. Comparing its root-line frame with the formal Cartier generator \widetilde y gives a frame in which this section is \widetilde y^\ell. Restricting to the Z_q gives compatible trivializations

Full faithfulness algebraizes this line-bundle trivialization and its section identity on Z; no algebraic map from Z to the root stack is needed. The generic fiber of Z therefore misses the positive part of D as well, and hence avoids all of D.

We next check the dimension assertion needed for the family. At a closed point z of F, flatness makes s a nonzerodivisor in \OO_{Z,z}, and its quotient is the Cohen–Macaulay ring \OO_{F,z} of dimension d. Thus \OO_{Z,z} is Cohen–Macaulay of dimension d+1. These local rings are also equidimensional: a Cohen–Macaulay local ring essentially of finite type over the excellent discrete valuation ring R has this property. Every component of Z meets F by properness, and no component is contained in F by flatness. The dimension formula now gives dimension d for every component of the generic fiber. Its finite image in X_{\C((s))} therefore has pure dimension d. After a suitable finite extension of \C((s)), choose a geometrically integral component of the reduced base change of this image.

A family dominating X. We have constructed compact subvarieties outside D. To obtain a dominating family, choose a positive component D_i with image dimension r, together with a component of S covering it. As the general chart and t vary, the images of points of F contain a dense open subset of D_i. We will show that each point in this open subset lies in the closed evaluation image of an algebraic family of d-dimensional subvarieties disjoint from D.

Now consider all Hilbert schemes of d-dimensional subvarieties of X. The geometrically integral loci parametrizing members disjoint from D have a countable stratification by integral parameter spaces with irreducible universal families. Denote the irreducible closures of their evaluation images in X by W_\alpha. Fix a point x in the dense open subset of D_i, and a point of a boundary fiber F mapping to it. Since Z has no vertical component, this point lies in the closure of a generic component. That component has dimension d by the preceding argument. Pass to a finite extension of the discrete valuation field over which the reduced generic components are geometrically integral, and let R' be the extended discrete valuation ring. Choose a generic component after this base change whose closure contains a point over the chosen point of F. Flatness still excludes vertical components, so such a choice is possible. Its reduced finite image is geometrically integral. Take the schematic closure of that image in X_{R'}. The structure sheaf of the closure has no uniformizer torsion, so the closure is flat over the discrete valuation ring and defines a morphism to the Hilbert scheme. Its generic member lies in one of the strata above. The corresponding W_\alpha is closed and contains the generic image, hence also its special-fiber support and the point x. Thus every point of the chosen open subset of D_i belongs to some W_\alpha. The generic component and the index \alpha may depend on x.

An irreducible variety over the uncountable field \C cannot have a dense open covered by countably many proper closed subsets. Some W_\alpha therefore contains D_i. By definition it also contains points outside D. Since D_i is a prime divisor in the integral variety X, the only proper irreducible closed subset containing it is D_i itself. Hence W_\alpha=X, and the family dominates X. On each member, avoidance of D makes the rational section s_0 regular and nowhere zero. It trivializes N_0, proving that L is numerically trivial on that member.□

Proof. When B is a point, every vertical divisor is zero. When the relative dimension is zero, the fiber bound makes f finite. Its geometrically integral connected generic fiber makes it birational, and normality of B then makes it an isomorphism. The assertion follows in both cases. We may therefore assume that the base dimension and the relative dimension are positive.

The fiber bound forces every vertical prime divisor to map onto a prime divisor of B. For each such prime P, write its full pullback as f^*P=\sum_jm_jE_j. Let b_j be the coefficient of G_Y along E_j, with zero coefficients included, and assign coefficient \min_j(b_j/m_j) to P in G_B. Only finitely many of these coefficients are nonzero. The residual divisor

is effective, vertical, \Q-Cartier, and relatively nef; over each base prime, at least one component is absent from its support. It remains to prove that R'=0.

Suppose R'\ne0, and choose a base prime under its support. Intersect B with general hyperplanes to obtain a complete-intersection curve meeting that prime at a general point; if \dim B=1, use B itself. Normal Bertini makes its inverse image in Y normal. This inverse image is integral: generic geometric integrality supplies the dominating component, and the fiber bound rules out further vertical components. Intersect upstairs with \dim Y-\dim B-1 general very ample hyperplanes. The result is a normal integral surface over the base curve, with geometrically connected generic fiber. Over the chosen general base point, these cuts retain curves both in the residual support and in a component missing from it. The curves remain distinct because the original components were distinct at the generic point of the base prime.

Resolve this surface. The pullback of the residual divisor stays effective, vertical, and relatively nef. It has zero intersection with the full fiber and nonnegative intersection with every fiber component. All those component intersections must therefore be zero. The fiber is connected, by generic connectedness and Stein factorization over the normal base curve. Its intersection matrix is negative semidefinite, with kernel generated by the full fiber. The residual part over the chosen point is thus a multiple of the full fiber. Its missing component forces that multiple to be zero, contradicting the retained component of its support.□

Completion of the proof of Theorem 3.1. The cases v=0 and v=n are covered by Lemma 3.2. For 0<v<n, Propositions 3.3, 4.1, and 5.1 give a dominating family of compact L-trivial subvarieties of dimension d=n-v.

Apply the nef-reduction theorem to a Cartier multiple of L [2, Theorem 2.1]. The resulting almost holomorphic rational map has connected fibers. The divisor L is numerically trivial on its compact general fibers and has positive degree on every noncontracted curve through a very general point of X. Write b for the base dimension. A compact L-trivial d-fold through such a point must be contracted: otherwise, general ample slices through the point would produce a noncontracted curve of L-degree zero. Consequently

To apply vertical descent, resolve the graph, flatten the main component over a modification of the base, resolve the base, and normalize the main transform. This gives projective morphisms

with \pi birational, B smooth, Y normal, and geometrically connected integral generic fiber of f. Every fiber has dimension at most n-b. Indeed, flatness and generic integrality keep the flat main transform integral after the base modification, and finite normalization preserves the bound on fiber dimension. This bound is the needed hypothesis; normalization need not preserve flatness.

We claim that \pi^*D has no horizontal component. Restrict to a very general fiber of f. There, \pi^*L is numerically trivial and \pi^*(L-cD) restricts to a pseudo-effective class. To justify the second assertion, choose a sequence of effective approximants with ample errors tending to zero. Shrink to a flat open of the base with geometrically integral fibers. This is possible because the generic fiber is geometrically integral. On this common open, the locus over which a given support contains the whole fiber is proper closed, by upper semicontinuity of fiber dimension. Avoid the countably many such bad closed loci in the base. Every approximant then restricts effectively to a very general fiber, and passage to the limit gives pseudo-effectivity of the restriction. The resulting class is -c\pi^*D. A negative nonzero effective divisor cannot be pseudo-effective on a projective variety, as its intersection with an ample power shows. The restriction of \pi^*D must therefore be zero. This also excludes b=0, which would force D=0 and contradict v>0.

It follows that G_Y=\pi^*G is vertical. It is also relatively nef because G_Y\sim_{\Q}\pi^*L. Lemma 5.2 now gives

for a \Q-divisor G_B on the smooth base. Every curve in B is dominated by a curve in Y, where the pullback has nonnegative degree; hence G_B is nef. Numerical dimension is unchanged by these pullbacks; the intersection formula gives

Together with b\le v, this yields b=v=\nu(B,G_B). Since G_B is nef on the b-dimensional base, it follows that G_B^b>0, so G_B is big. Pullback of sections then gives \kappa(X,L)\ge b=v. The inequality \kappa(X,L)\le\nu(X,L) for nef divisors provides the reverse bound and proves abundance.□

Proof. If q(X)>0, resolve the Stein factorization of the Albanese map to obtain a morphism f:W\to B with connected fibers, with W,B smooth projective and \dim B>0. The map from B to a subvariety of \Alb(X) is generically finite. Wedges of invariant one-forms on the abelian variety therefore give a nonzero section of K_B: at the generic point, choose \dim B independent pulled-back one-forms. Thus \kappa(B,K_B)\geq0.

For a very general smooth fiber F, the restriction of the pseudo-effective class K_W is pseudo-effective. For example, restrict a positive current representing it to almost every fiber, or use effective approximations with arbitrarily small ample error. Adjunction identifies this restriction with K_F. If F has positive dimension, the induction hypothesis gives \kappa(F,K_F)\geq0; for a point the same assertion holds by convention. Applying Theorem 1.2 with both boundaries empty gives \kappa(W,K_W)\geq0, contradicting (10).

This is the only use of Theorem 1.2 in the proof. Irregularity is a smooth birational invariant. Finally, when \Pic^0=0, a numerically trivial line bundle is torsion, since the group of numerically trivial line bundles modulo \Pic^0 is finite. Clear denominators for rational divisors. On a terminal \Q-factorial model, pull back to a smooth resolution and then descend the rational linear equivalence.□

Proof. Shrink to an open U where the evaluation has finitely many etale sheets, the parameter map is smooth, and the generated involutive distribution \mathcal F has constant rank. Its construction is algebraic: spans and brackets stabilize at the generic point after finitely many operations, and descend from the etale sheets. The nonempty open parts of the integral parameter fibers are connected. A chain step moves between two points in the image of one such open fiber. We use only steps lying in U, so each step remains in one analytic leaf of \mathcal F.

For x\in U, endpoints of chains of at most k steps form a constructible set, by the algebraic fiber-product construction and Chevalley’s theorem. Every endpoint lies in the analytic leaf through x. A locally closed smooth variety contained in an analytic leaf has dimension at most \rk\mathcal F. Indeed, in a Frobenius chart the leaf meets at most countably many plaques: use finite plaque chains in a countable foliation atlas. The transverse coordinate map on any connected smooth piece then has countable image and is constant.

There is therefore a maximal dimension among the closures of all such endpoint loci; it is attained for some finite k. The loci are nested because we allow at most k steps, including the identity step. Choose an irreducible component V of maximal dimension and a dense open O\subset V of reachable points. On each etale sheet, the parameter map defines the algebraic relation consisting of pairs in the same fiber. Restrict its first coordinate to the inverse image of O, take the component through the diagonal, and map the second coordinate to U. The closure of this image contains O, hence V, while its points are reachable in at most k+1 steps. Maximality forces that closure to equal V. At a general diagonal point, smoothness of the parameter map identifies its vertical tangent with the corresponding web tangent. Every web tangent is consequently tangent to V, and so is every bracket. Hence \dim V\geq\rk\mathcal F, while the reverse inequality was proved above. A plaque is thus open in V. The equations of V, pulled back to the connected immersed leaf, vanish on a nonempty open and hence on the entire leaf. Its closure is exactly V.

These invariant subvarieties of dimension \rk\mathcal F have countably many Hilbert or Chow parameter spaces. Tangency on the regular locus is an algebraic condition after stratification, so one such family dominates. An invariant irreducible variety meeting U contains the leaf through any of its points in U: the tangent vector fields preserve its reduced ideal, first on its smooth locus and then everywhere by closure. A leaf closure of dimension \rk\mathcal F is therefore unique through a general point. Assigning that closure gives the desired rational map. Resolve its graph and take the Stein factorization.□

Proof. Otherwise take a covering family of nongeneral-type subvarieties, with smooth domains V, and slice its parameters as above. Restricting the ramification formula of its generically finite total evaluation shows that K_V dominates the restriction of the pulled-back K_X by an effective divisor. Thus K_V is pseudo-effective. The induction hypothesis gives a good minimal model of V, so \kappa(V)\geq0. Since V is not of general type, its general Iitaka fibers have positive dimension and Kodaira dimension zero. Spreading these fibers gives another covering family with smooth domains G, of dimension less than n, and \kappa(G)=0. Each G has a good minimal model and \kappa_\sigma(K_G)=0.

To exploit this zero-Iitaka-dimension family, fix a positive current T representing K_X. After slicing and resolving the total evaluation of the G’s, its restriction to almost every parameter fiber, plus the effective restricted ramification divisor, is a positive current in K_G. Every positive current in that class is the current of the fixed canonical divisor of G. To see this, let R be such a current and take a common resolution u:H\to G, v:H\to G_{\min} with the good minimal model. The current

is positive and represents K_H. Since K_{G_{\min}} is torsion, this class is represented by an effective v-exceptional divisor. Its intersection with the (\dim G-1)-st power of an ample pullback from G_{\min} is zero. That form is strictly positive off the exceptional locus, so the current is supported there. The support theorem makes it divisorial, and independence of exceptional divisor classes, by negativity, determines its coefficients. Pushing forward by u determines R itself. After shrinking the parameter space, these fixed canonical divisors are the fixed divisor of a sufficiently divisible relative pluricanonical system. Remove this divisor and the excluded loci of the construction from the total family. Their images under generically finite evaluation are proper algebraic subsets of X.

It follows that T annihilates the web distribution of the G’s on a dense open. If that distribution has full rank, T is supported on a proper algebraic set. The support theorem for positive closed (1,1)-currents expresses it as a finite nonnegative real combination of prime divisors [58, 17]. Its rational cohomology class then has a nonnegative rational representative: solve the finite rational linear system for the coefficients on the same face of the nonnegative orthant. Lemma 6.1 converts numerical effectivity to \Q-linear effectivity, a contradiction.

Otherwise Lemma 6.2 gives a rational quotient with general smooth fiber F on a smooth resolved graph, where 0<\dim F<n. Its canonical divisor is pseudo-effective: restrict the pseudo-effective canonical class of the smooth graph to a very general fiber, as in Lemma 6.1. The moving G’s still generate its tangent space at general points. Indeed the quotient is constant on every general web member, hence induces a rational map on their parameter space; restricting to a general quotient fiber preserves dominance of evaluation. A pluricanonical rational map of F is constant along each such G. Slice the parameters inside F to make the total evaluation generically finite: ramification injects the restricted pluricanonical sections into sections of a multiple of K_G, and these span a space of dimension at most one. Fix a nonzero denominator section and work where it does not vanish. Its restriction is nonzero on a general member of the covering family, so all ratios are constant on that member. Their differentials vanish on the web and its brackets, so the pluricanonical map is constant on F. Lower-dimensional nonvanishing therefore gives \kappa(F)=0, and its good minimal model gives \kappa_\sigma(K_F)=0.

The quotient has now reached the exact setting of the numerical-zero-fiber reduction of Gongyo–Lehmann [31, Theorem 1.3 and Corollary 4.5]. For a projective \Q-factorial klt rational pair with a connected-fiber morphism and numerical dimension zero on the general fiber, that theorem produces a klt pair on a smooth birational base whose good-minimal-model existence is equivalent. Here the total space is the smooth graph, the boundary is zero, and the base has dimension less than n. The required numerical dimension is \kappa_\sigma, which is zero by the good model of F; thus the numerical-dimension qualification, with the corrected convention discussed in [23, Section 3 and Theorem 3.2], causes no issue. The induction hypothesis supplies the base good model, hence nonvanishing upstairs, contradicting (10).□

Proof. Resolve a moving subsystem of a multiple of P. If its map is generically finite, P is big and the assertion follows from pseudoeffectivity of K_Y. Otherwise its general fiber is a proper positive-dimensional subvariety through very general points and is of general type by Proposition 6.3. The canonical divisor of the smooth resolution is relatively big, so adding a sufficiently large ample pullback from the image makes it big. Since that pullback is bounded by a multiple of the resolved moving subsystem, pushforward gives K_Y+aP big for some a>0. Convexity with the pseudo-effective K_Y, and then addition of the pseudo-effective P, gives the assertion for every c>0. A nonzero nonnegative functional on a closed convex cone is strictly positive on its interior. Applying it to K_Y+cP proves the last assertion.□

Proof. We induct on the integer part of M; discrepancies over a smooth variety are positive integers, so the assertion is empty when M<1. Suppose that a strictly increasing sequence exists, and discard initial terms so its members exceed a fixed positive number c. Skoda integrability and change of variables give

at a general point x of the center. Indeed every exponent smaller than 1/\nu_x(T) is locally integrable, whereas integrability after pullback imposes the corresponding discrepancy bound. Hence all centers lie in the fixed proper Siu locus \{\nu_x(T)\geq c/M\}, which is algebraic by Siu analyticity and projectivity [58]. Passing to a subsequence, all centers lie in one irreducible component of this locus.

If, after passage to a subsequence, the centers lie in a fixed codimension-at-least-two subvariety, principalize its ideal. All lifted centers lie in the exceptional locus, where the relative canonical divisor has positive integral order. The discrepancy bound for the new smooth ambient space has therefore decreased by at least one. Pull back T and apply induction.

Otherwise the centers lie in a fixed prime divisor D of the Siu locus. Write T=c_D[D]+T' with T' positive and with zero generic Lelong number along D. The integers \ord_E(D) are bounded: the fixed effective divisor D has positive log canonical threshold, and \operatorname{lct}(D)\ord_E(D)\leq A_V(E). Pass to a constant order. The residual numbers \nu_E(T') are still strictly increasing, and after discarding a term have a positive lower bound. Equation (11) places their centers in a fixed Siu locus of T', which does not contain D. Their intersection with D has codimension at least two, so the preceding case applies.□

Proof. Fix a current T as in the statement and suppose such a fibration exists. Consider all dominant rational maps from Y with connected general fiber of positive dimension for which the pulled-back current annihilates the fiber tangents on a dense regular open of a resolved graph. Choose one with smallest base dimension b. We will descend the current to this base and find a covering family of curves on which the descended current vanishes. Their quotient will give a member of the same collection with smaller base dimension. If b=0, the current vanishes on a dense open; the divisorial-current argument in Proposition 6.3 contradicts (10). Thus 0<b<n. Resolve the graph and the space carrying the current:

Here W,B are smooth projective and h has connected general fiber. All further modifications replace this graph and base by birational models; they do not change the chosen rational map or the function field of its base. Flatten the main component over a modification of the base, resolve that base, normalize the main transform, and then resolve upstairs. Every vertical prime on the flat transform maps to a divisor of the base: a prime over base codimension c would have generic fiber dimension at least n-b+c-1, whereas all fibers have dimension at most n-b. A prime of W either maps onto a prime of the flat transform, where this dimension count applies, or is exceptional over that transform. In the latter case it is also exceptional over Y, because the transform maps birationally to Y. Thus every vertical divisor on W over base codimension at least two is exceptional over Y. This remains true after later resolutions and base modifications: a prime dominating a divisor of Y already has divisorial center on the old base, and its center on a higher base model is the strict transform of that divisor.

For divisors and numerical classes put

This strict pullback is well defined on numerical classes because Y is \Q-factorial: numerical classes upstairs decompose into pulled-back classes from Y and exceptional classes. It preserves pseudoeffectivity. For a higher base model r:B'\to B, its analogue satisfies T_{B'}r^*=T_B, by computing on a common graph. The operator T_{B'} kills divisors exceptional over B: otherwise a prime in such a pullback that dominates a divisor of Y would have center of codimension at least two on B, contrary to the property just established.

Descent of the current. Let T also denote the pulled-back current on W. Shrink to a smooth open B^0 so that h is smooth proper, the removed set contains no vertical divisor over B^0, and the remaining open in every fiber is nonempty and connected. Away from that fixed removed algebraic set upstairs, T descends to a positive current S^0. Indeed in submersion coordinates the horizontal coefficient distributions are independent of fiber variables by closedness, and the open parts of general fibers are connected. They therefore glue to S^0 on B^0. The difference T-h^*S^0 is closed of order zero and supported on the removed algebraic set. The support theorem expresses it as a signed divisor sum; a closed order-zero (1,1)-current supported in codimension at least two is zero. Its horizontal coefficients are nonnegative, since a pullback has zero generic Lelong number along a horizontal divisor.

Subtract those finitely many global Siu components from T, obtaining a positive current T' that equals h^*S^0 on all of h^{-1}(B^0). Choose a Kähler form \omega representing an ample algebraic class and put d=n-b. Fiber integration gives h_*(\omega^d)=c>0, constant on B^0, by closedness. The projection formula therefore gives

The left side defines a global positive closed extension. Its class is real algebraic, since \{T'\} is the algebraic class p^*K_Y minus real divisor classes and algebraic intersection and pushforward preserve such classes. Remove its divisorial parts along B\setminus B^0, and call the result S. On a dense open T=h^*S. Their difference is again closed of order zero, so the same support theorem leaves only a signed divisor sum. Over B^0, this difference consists only of the horizontal components removed above. Its coefficient at a prime dominating a base prime that meets B^0 is therefore zero. Along a base prime contained in B\setminus B^0, the current S has zero generic Lelong number by construction. At a prime upstairs dominating it, the generic local map is a transverse power map times a submersion. Local target balls of a radius equal to a fixed power of the source radius give the Lelong comparison, so h^*S also has zero generic number there. Positivity of T then makes the coefficient of T-h^*S nonnegative. Thus all coefficients at primes dominating base primes are nonnegative. The only possible negative coefficients lie over base codimension at least two and are exceptional over Y. We have therefore obtained a positive closed current S in a real algebraic class on B, with T-h^*S a divisor current whose only possible negative coefficients are p-exceptional. Push this difference forward by p. The exceptional terms disappear and the remaining terms are effective, so

The horizontal components removed during descent contribute effective divisors to the same difference.

A negative canonical direction on the base. Choose a nonzero supporting functional \alpha of the pseudo-effective cone at K_Y, and put \eta=T_B^*\alpha. The corresponding functionals \eta' on higher smooth base models are movable classes by pseudo-effective/movable duality [8, Theorem 0.2]. They annihilate base-exceptional divisors. Since S is positive, \eta\cdot\{S\}\geq0. Applying \alpha to (13) gives the reverse inequality, because \alpha(K_Y)=0. Hence \eta\cdot\{S\}=0. If c_D[D] is a positive Siu component of S, both S-c_D[D] and [D] are positive. It follows that \eta\cdot D=0, or equivalently \alpha(T_BD)=0.

Before applying multiplier approximation, we need to control the divisorial part of this descended current. Only finitely many prime divisors on B have positive divisorial Lelong coefficient for S. Otherwise their nonzero strict pullbacks are effective divisors on Y killed by \alpha, with no common prime components. Indeed, a nonexceptional prime of Y cannot map into the intersection of two base primes. Only finitely many base primes have zero strict pullback, since their total pullbacks are supported in the finite exceptional locus of p. In the finite-dimensional rational space of divisor classes, infinitely many remaining strict pullbacks have a rational relation. This relation must have both positive and negative coefficients: a nonzero effective sum has positive intersection with an ample (n-1)-fold product on Y, so cannot be numerically zero. The two sides are thus nonzero effective rational divisors with no common prime component. Lemma 6.1 makes them \Q-linearly equivalent. Clearing denominators gives two distinct linearly equivalent effective integral divisors and hence a pencil, whose class is killed by \alpha. This contradicts Corollary 6.4.

We claim that

The smooth general fiber F of h is of general type by Proposition 6.3. The relative-positivity theorem of Kovács–Patakfalvi [44, Theorem 9.9] applies to a log canonical fiber space with smooth base, SNC total pair, and log-general-type geometric generic fiber; for a rational base divisor M with \kappa(M)\geq0, it gives relative subadditivity with the term h^*M. Here both spaces are smooth projective, the boundary is zero, and we take M=0. Invariance of plurigenera on the smooth locus identifies the generic-fiber Kodaira dimension with that of F. Thus

This is an established general-type-fiber theorem, not another use or strengthening of Theorem 1.2. Choose compatible canonical divisors and set P=K_Y-T_B(K_B)=p_*(K_W-h^*K_B). For every sufficiently divisible m, pushing forward an effective divisor \operatorname{div}_W(\varphi)+m(K_W-h^*K_B) gives \operatorname{div}_Y(\varphi)+mP\geq0. The resulting injection of sections preserves their ratios, so \kappa(Y,P)\geq1. Corollary 6.4 now gives 0<\alpha(P)=-K_B\cdot\eta, proving (14).

Rational curves and an exact zero gap. Apply the multiplier approximation to S on B, and principalize \mathcal J(kS) by r_k:B_k\to B. If F_k is its divisor, the class

is nef. Put \eta_k=T_{B_k}^*\alpha. The class \eta_k is movable and F_k is effective, so 0\leq P_k\cdot\eta_k\leq (L_k\cdot\eta)/k. Since \eta\cdot\{S\}=0 and L_k-k\{S\} stays in a bounded set, this gives the first estimate below. The second uses the fact that \eta_k kills exceptional divisors:

Fix an ample divisor H on B, and write \delta=-K_B\cdot\eta>0. On B_k, choose an ample divisor H_k and a sufficiently small rational \epsilon_k>0 so that the ample rational class

satisfies Q_k\cdot\eta_k=O(k^{-1}). This is possible because \eta_k\cdot r_k^*H=\eta\cdot H is fixed. Approximate \eta_k by a positive sum \gamma_k=\sum_{i=1}^{N_k} a_i[C_i], with a_i>0, of covering-curve classes, closely enough that

for a constant C independent of k. Keep only terms with K_{B_k}\cdot C_i<0. Their total anticanonical degree is at least \delta/2, while their total Q_k-degree is at most C/k. Taking the weighted average of the ratios therefore gives one covering curve \Gamma_k with

The quantitative bend-and-break estimate [51, Theorem 5] gives a rational curve through a general point of \Gamma_k with Q_k-degree at most 2\dim B times this ratio. Parameterization and uncountability give a covering family of such rational curves, denoted R_k. Since Q_k-P_k-k^{-1}r_k^*H is ample and both P_k and r_k^*H are nef, we obtain

Shrink each parameter space so that the curve class and its contacts with the exceptional divisors and the strict transforms of the Siu divisors of S are constant. The general parametrized member is free in characteristic zero. Consequently K_{B_k}\cdot R_k\leq-2. Put \bar R_k=r_{k*}R_k. Its bounded H-degree bounds K_B\cdot\bar R_k, and effectivity of the relative canonical divisor gives

In particular, the upper bound is independent of k. If an exceptional prime E has coefficient a_E=A_B(E)-1 in K_{B_k/B}, a positive contact with it contributes a_E(E\cdot R_k) to this bounded sum. Both factors are positive integers. Thus the number of exceptional primes met by R_k, their log discrepancies A_B(E)=a_E+1, and their contact degrees are uniformly bounded. Contacts with the finitely many strict transforms of divisorial Lelong components of S are bounded by their fixed degrees against \bar R_k. Subtracting all these divisorial parts from r_k^*S leaves a positive current. Its restriction to almost every member of the family is defined and positive, by the local-potential and Fubini argument following Lemma 6.2. Its degree is the first difference below; this proves that difference is nonnegative. Using (12) and (16) gives

The sum includes exceptional primes and those strict divisorial components; terms with zero contact are irrelevant. For the last bound, the expression involving F_k equals P_k\cdot R_k+(\{S\}-L_k/k)\cdot\bar R_k. The first term is O(k^{-1}) by (16); the second has the same bound because L_k-k\{S\} stays bounded and the H-degrees of \bar R_k are bounded.

Bounded ample degree gives only finitely many numerical classes of the integral cycles \bar R_k. Pass to one class, so \{S\}\cdot\bar R_k is now one fixed real number. The current S on B also remains fixed. By Lemma 6.5, the sums in (18) belong to an ACC set: there are a bounded number of terms, their nonnegative integral multiplicities are bounded, and all relevant discrepancies are bounded. Finite sums of this kind preserve ACC, by successively taking nonincreasing subsequences of their entries. We can now replace the limiting gap by an exact equality: the nonnegative gaps in (18) must equal zero for some covering families. Otherwise a sequence tending to zero has a strictly decreasing positive subsequence, giving a strictly increasing sequence of the complementary sums.

For a family with zero gap, restrict the residual current to almost every parameter fiber as above. Its integral on that complete curve is zero. A positive current on a curve is a positive measure, so zero integral forces it to vanish. Off the removed divisors the residual current agrees with r_k^*S. The discussion following Lemma 6.2 and that lemma itself yield a positive-relative-dimensional rational quotient of B whose vertical directions are annihilated by S on an open. Composing with h and taking the Stein factorization gives a quotient of smaller base dimension whose vertical directions are annihilated by T. It belongs to the collection minimized at the start, contradicting the choice of b.□

Proof. Pass to a common smooth model V also dominating the fixed resolution carrying T. Choose on the fixed resolution a sufficiently positive line bundle H. For sufficiently divisible k, multiplier approximation gives a globally generated \OO(kp^*K_Y+H)\otimes\mathcal J(kT). Pull this system to V. For the finite collection of primes on the common resolution whose coefficients we are comparing, choose a general member D_k of the underlying effective divisor system. At every such prime E, it has multiplicity \ord_E\mathcal J(kT); global generation after removal of the ideal ensures no additional generic vanishing. Here the notation H also denotes its pullback to V.

Put D'_k=p_{t*}D_k and H_t=p_{t*}H. The first is effective, and both are \Q-Cartier because Y_t is \Q-factorial. Pushing the rational linear relation gives

Thus the exceptional divisor D_k-p_t^*D'_k is rationally equivalent to

The two exceptional divisors are equal: their difference is numerically trivial and exceptional, so the negativity lemma applied with both signs makes it zero. Since p_t^*D'_k is effective,

The last coefficient is fixed for this model and t. Combining with (12), dividing by k, and letting k\to\infty proves the assertion. The same proof can be made on each common model, so the comparison applies to all divisorial valuations needed later. No uniformity in t of the fixed error is required.□

Proof. Suppose such families exist. Fix once and for all a smooth resolution p:W\to Y and a positive current T in the class p^*K_Y; pseudoeffectivity supplies this current. Neither W nor T will depend on t. Let g_0 be a common upper bound for the genera, and choose an integer m>0 such that mK_Y is Cartier.

We first compare the curves on these fixed models without changing their domains. Slice the parameter space so that the smooth total family has dimension n, the parameter space has dimension n-1, and evaluation is generically finite. It has rational maps to Y, to W, and to the common models comparing Y and Y_t. Each target is proper, so each map extends at codimension-one points by the valuative criterion. Its indeterminacy locus has dimension at most n-2, and its image cannot dominate the parameter space. Remove these finitely many images from that space, then resolve and compactify away from the remaining complete fibers. Denote the resulting maps by \rho_t:U_t\to W and q_t=p\circ\rho_t:U_t\to Y. The smooth general parameter fiber is still the original C_t, with the same genus. The current we use on this varying space is always \rho_t^*T.

For a prime Z of U_t exceptional over Y, the restricted valuation of its function field is r_Z\ord_E for a divisorial valuation E over Y. The coefficient of Z in the ramification difference is

This follows by factoring through a model extracting E and calculating the tame ramification of DVRs. It is positive and bounded away from zero: Y is terminal, so a(E;Y,0)>1, and the index m puts these log discrepancies in m^{-1}\Z. In particular a(E;Y,0)\geq1+1/m. The remaining ramification coefficients are nonnegative. On the general parameter fiber,

The latter inequality follows from pseudoeffectivity and the covering property. Thus

Each exceptional contact contributes (r_Za(E;Y,0)-1)(Z\cdot C_t) to this bounded sum. The first factor is at least 1/m, and the second is a positive integer. Formula (20) therefore bounds the number of contacted exceptional primes, their log discrepancies, their ramification indices, and their contact degrees. In particular the integers r_Z(Z\cdot C_t) are uniformly bounded. Also

so these degrees take only finitely many values.

The ramification estimate has bounded the possible contacts. We next compare their Lelong contributions with the canonical differences. The generic Lelong number of \rho_t^*T at Z is r_Z\nu_E(T). Extract E, remove its generic divisorial part, and use the transverse-power-map comparison for the zero-generic-number remainder. Lemma 6.7 shows that these coefficients dominate the coefficients of the pulled-back canonical difference J_t. Its support is exceptional over Y as well as Y_t, since the map between these models is small. Subtracting all exceptional divisorial parts of \rho_t^*T leaves a positive current. Shrink the parameter space so that the relevant divisor contacts are constant, and restrict this residual current to almost every fiber. As in (18), its degree is the first difference below and is nonnegative. Consequently

All divisor contacts used here are nonnegative, since the curves cover the total space.

On the fixed smooth resolution p:W\to Y, one has

because the relative canonical divisor is effective. Thus Lemma 6.5 applies to the one fixed current T on W, with a fixed discrepancy bound. The sums in (21) form an ACC set: their Lelong numbers belong to this ACC set, and both the number of terms and their integral multiplicities r_Z(Z\cdot C_t) are bounded. Pass to a fixed value of q_t^*K_Y\cdot C_t. Since \mu_t\to0, the exact-zero argument of (18) gives a family for which the left gap is zero.

For that family, the restriction of the residual positive current to almost every parameter fiber is a positive measure of total mass zero, so it vanishes. On the dense open away from the removed exceptional divisors it agrees with \rho_t^*T. Apply Lemma 6.2 to the family evaluated by \rho_t:U_t\to W. Its curve tangents generate a rational fibration of positive relative dimension on the smooth projective variety W, and T annihilates the fiber tangents on a dense open. Transport this rational fibration through p:W\to Y on their common isomorphism open. This contradicts Proposition 6.6.□

Proof. In an infinite ample-scaling program the scaling limit is zero: a positive limit is covered by the termination theorem for a dlt pair with an ample summand in the scaling divisor [5, Theorem 1.9(ii), equivalently Theorem 4.1(ii)]. Discard the finitely many divisorial contractions and write the flips as V_i\dashrightarrow V_{i+1}/Z_i, with scaling numbers \lambda_i>0 tending to zero. Fix a component S_1 of \lfloor\Delta_1\rfloor, write S_i for its normal strict transform, and let T_i be the normalization of its image in Z_i. Since the ambient contraction is small, S_i\to T_i is projective birational. The standard discrepancy argument makes S_i\dashrightarrow S_{i+1} an isomorphism in codimension one after finitely many steps [3, proof of Lemma 3.6].

To identify the lower-dimensional input, consider the relative program induced on this one component of the floor. Take a small projective \Q-factorialization p_i:S'_i\to S_i, and put

Adjunction gives a dlt pair (S'_i,B_i), with B_i,C_i\geq0, and (S'_i,B_i+\lambda_i C_i) is lc. The scaling divisor has no floor component, so its restriction is effective. The number \lambda_i is rational, being the ratio of intersections of rational divisors on the contracted rational curve. The globally nef divisor L_i=K_{V_i}+\Delta_i+\lambda_i A_i descends rationally linearly across V_i\to Z_i. Indeed, for sufficiently small rational \delta>0, the pair (V_i,(1-\delta)\Delta_i) is klt and its negative adjoint is ample over Z_i. Relative basepoint freeness applies to a Cartier multiple of L_i; its numerical triviality and the connected fibers give descent. Restricting and pulling back yields

The descended divisor on T_i is nef: its pullback to S'_i is the pullback of the restriction of the globally nef L_i.

By [5, Theorem 1.1(3)], a D_i-LMMP over T_i with scaling of a fresh ample divisor terminates. The theorem applies to the effective rational lc boundary B_i+\lambda_i C_i, with \lambda_i C_i rational Cartier and the driving pair \Q-factorial dlt. Since the base map is birational, the endpoint is a log minimal model, not a Mori fiber space. Comparison with the relatively ample model S_{i+1} identifies this endpoint with a small \Q-factorialization S'_{i+1}, as in [5, Remarks 2.9–2.10].

The next point is that these finite relative programs concatenate into an absolute program with scaling of the transforms of C_1. Throughout the i-th piece, D_i+\lambda_i C_i remains the pullback of a nef divisor on T_i, hence is globally nef. Here D_i and C_i also denote their transforms along this relative program. A contracted negative ray R satisfies

It follows that C_i\cdot R>0. Every smaller coefficient of C_i leaves this ray negative, while D_i+\lambda_i C_i is globally nef. Thus the absolute scaling threshold at that step is exactly \lambda_i. The relative cone is a face of the absolute cone, cut out by the pullback of an ample divisor on the projective base T_i; its extremal rays are therefore absolute extremal rays. Thus the pieces concatenate as asserted in the cited remarks. Rescaling the initial scaling divisor by \lambda_1, if necessary, makes its sum with the initial boundary lc and nef.

If the concatenation were infinite, its positive scaling numbers would tend to zero. Fix a late scaling value \lambda_j. On every earlier ray R contracted at threshold \lambda_i, the calculation above gives

Thus all preceding steps are nonpositive for the adjoint with this fixed coefficient \lambda_j. On a common resolution of the initial and the j-th models, the pullback of D_1+\lambda_j C_1 equals the pullback of the late nef divisor D_j+\lambda_j C_j plus an effective comparison divisor. Push this equality down to the initial model. The pushforward of the nef pullback is pseudo-effective, as is the effective comparison term, so D_1+\lambda_j C_1 is pseudo-effective. Closedness of the pseudo-effective cone and \lambda_j\to0 show that D_1 is pseudo-effective. Assumption 2.1 therefore gives an absolute log minimal model for that initial pair. The concatenation terminates by [5, Theorem 1.9(iii)], since its zero limit is never attained. The floor-component argument of [3, Lemma 3.6] now gives special termination. The only good-model input in this argument concerns the absolute pair on the lower-dimensional floor component. In particular it does not require an effective representative in dimension n.

For the final assertion, a curve disjoint from the round-down has degree zero against any signed divisor supported there. It therefore cannot generate an adjoint-negative flipping ray. Special termination rules out all sufficiently late flips. There are only finitely many divisorial contractions, since each lowers the Picard number, and pseudo-effectivity excludes a Mori fiber space as the endpoint. Thus the endpoint is nef.□

Proof. Put P=K_W+D, and suppose it is not big. It cannot have Iitaka dimension at least one: by Corollary 6.4, K_W+cP would be big, and (1+c)P=(K_W+cP)+D would then be big as well. Hence \kappa(W,P)\leq0.

Run the dlt LMMP for P with ample scaling. The divisor P has a signed rational representative supported on the floor D, and its transforms retain that property. Every negative flip must meet the floor: on its complement the representing rational section is nowhere vanishing, so the adjoint has zero degree on every complete curve there. By Proposition 6.9, an infinite sequence would eventually have all flips disjoint from the floor, a contradiction. Divisorial contractions are finite in number. Thus the program terminates at a \Q-factorial dlt log minimal model (Z,D_Z).

The boundary is reduced, K_Z is pseudo-effective as the birational pushforward of K_W, and the nef adjoint has a signed rational representative supported on D_Z. Its restriction to D_Z is semiample by Proposition 2.3. Apply Theorem 3.1 with L=K_Z+D_Z and c=1, so that the required pseudo-effective class L-cD_Z is K_Z. It gives \kappa(Z,L)=\nu(Z,L)\geq0. The Iitaka dimension is at most zero, as proved above, so both dimensions are zero. Intersecting K_Z+D_Z\equiv0 with an ample (n-1)-fold product, and using pseudoeffectivity of K_Z, forces D_Z=0. The signed relation then gives K_Z\sim_\Q0.

The signed representative of K_W is supported on D: pull back the given representative of K_t and add the relative canonical divisor. Since D_Z=0, every component of this support is contracted over Z. On a common resolution a:V\to W, b:V\to Z, we therefore have a^*K_W\sim_\Q E for a signed b-exceptional divisor E. The class a^*K_W is pseudo-effective. A positive current in this class has zero intersection with the (n-1)-st power of an ample pullback from Z, so is supported on the exceptional locus. The support theorem makes it effective divisorial. Independence of exceptional divisor classes, by negativity, identifies its coefficients with those of E; hence E\geq0. Pushing forward by a shows that K_W is \Q-linearly effective, contradicting (10).□

Proof. Suppose such a family exists, and denote its birational maps by \phi_a:T_1\dashrightarrow T_2. Choose one map \phi_{a_0} in this family and use it to identify T_2 birationally with T_1. The maps g_a=\phi_{a_0}^{-1}\circ\phi_a are then birational selfmaps of T_1. The evaluation (a,u)\mapsto(u,g_a(u)) is dominant onto T_1\times T_1; in particular, the images of a general point under these selfmaps are dense.

We use the birational automorphism group to identify this fixed cover. The cover is non-uniruled, being generically finite over the non-uniruled Y. Hanamura’s non-uniruled birational-group theorem gives a smooth projective birational model for which the reduced birational group is a group scheme, locally of finite type, and its identity component is an abelian variety of regular automorphisms [35, Theorems 2.1–2.2]; see also [7, Proposition 3.7 and Theorems 3.8–3.9]. After conjugating the maps to that model, shrink the irreducible parameter variety anew so that their graph closures are flat with integral fibers and both graph projections remain birational on every fiber. This is a family in the flat graph functor of [7, Definition 3.4], which is represented by the birational scheme [7, Proposition 3.7]. It therefore defines a morphism to that scheme; because the parameter variety is reduced, this morphism factors through its reduction. Its connected image lies in a single component, hence in a translate of the identity component. Translate the family by one member so that it lies in that identity component; this preserves dominance of evaluation. Thus the identity component, an abelian variety, has a dense orbit. That orbit is the quotient by the stabilizer of a general point. A quotient of an abelian variety is again an abelian variety, so the fixed cover is birational to an abelian variety.

It remains to see why this abelian cover contradicts the original counterexample. Resolve the generically finite rational map from that abelian variety to X: on a smooth model U, ramification gives

The canonical class of U also has the effective representative E_A exceptional over the abelian variety: choosing a nowhere-zero top form on that variety gives K_U\sim E_A. This second choice of representative need not be the canonical divisor compatible with the displayed ramification formula. Let T be a positive current in K_X. The current f^*T+[R] is positive and represents K_U. Let \alpha be an ample class on the abelian variety. The intersection of K_U with the (n-1)-st power of its pullback is zero, since K_U\sim E_A and E_A is exceptional. The positive current f^*T+[R] therefore has zero mass against that power. On the open where the birational map to the abelian variety is an isomorphism, the pulled-back ample form is strictly positive, so the current vanishes there. Its support is consequently exceptional over the abelian variety. The positive summand f^*T is supported there as well, and the support theorem makes it divisorial. Push forward this summand alone: f_*(f^*T) is a positive divisorial current whose class is \deg(f)c_1(K_X) by the projection formula. Rationality of the numerical class supplies a rational effective representative, and irregularity zero turns numerical effectivity into nonvanishing, as in Lemma 6.1. This contradicts \kappa(X,K_X)=-\infty.□

Proof. Apply the dlt blowup theorem for an effective rational divisor [29, Theorem 2.10]. It gives a projective birational morphism p:Q\to Z with Q \Q-factorial and

where (Q,B_Q) is dlt and E_Q\geq0 is the surplus over the non-lc locus. In this construction the coefficients of the strict boundary are truncated at one and the relevant extracted divisors are reduced; the coefficients left over form E_Q. The surplus is absent over the open where (Z,\Theta) is lc. In particular it is absent over a neighborhood of the generic point of V.

There are lc strata of (Q,B_Q) mapping onto V. Indeed over that lc neighborhood the construction is crepant, and the center of a log-discrepancy-zero divisor on the dlt model is an lc stratum. Choose a stratum S minimal among those mapping onto V. Iterated dlt adjunction makes S normal and gives an effective rational dlt different on S; its lc centers are precisely the smaller ambient strata in S [42]. The divisor E_Q does not contain S. Since Q is \Q-factorial, a Cartier multiple of E_Q restricts to an effective Cartier divisor on S. Adding that restriction to the different gives an effective rational divisor B_S such that

Over the generic point of V this pair is klt: the surplus is absent there, and a non-klt center of the different dominating V would be a smaller stratum, contradicting the choice of S.

Factor the map through the normalization V^\nu and its Stein factorization:

Here q has connected fibers and \eta is finite. Put D_V=\nu^*((K_Z+\Theta)|_V) and D'=\eta^*D_V. We apply the canonical bundle formula on the connected-fiber base V', then push the resulting intersection inequality through the finite map \eta. Its ramification will contribute an effective term. The final comparison is between the canonical cycle on V^\nu and the canonical divisor on V^*.

We verify the klt-trivial-fibration hypotheses for q:(S,B_S)\to V'. The generic pair is sub-klt, and Equation (24) is the required rational-linear pullback expression. For the rank condition, take a log resolution over the generic point of V'. Since the boundary on S is effective and the generic pair is klt, the round-up of its discrepancy divisor has coefficient zero along every strict boundary component and has nonnegative coefficients only on exceptional divisors. Over this dense klt open, its pushforward to the normal S is \cO_S: a rational function allowed poles only over exceptional centers extends across codimension at least two. Connectedness in the Stein factorization therefore gives generic rank one for the required discrepancy round-up pushforward. This verifies the rank condition, rather than deducing it from connected fibers alone.

The canonical bundle formula, with Ambro’s moduli positivity in the form recalled by Fujino–Gongyo, now gives

where M is a b-nef moduli divisor [27, Definition 3.1 and Theorem 3.5]. The definition requires sub-klt over the generic point; it does not require the pair to be globally sub-lc. Thus the possible non-lc fibers caused by E_Q|_S are permitted.

The discriminant B_{V'} on this initial base is effective. To check this at a prime divisor F\subset V', work over its generic point, where the base is regular. A component of its inverse image on S has multiplicity m_F\geq1 and nonnegative boundary coefficient b_F. Its discrepancy bounds the generic lc threshold t_F from above by

Thus the discriminant coefficient 1-t_F is nonnegative. If the fiber is already non-lc, t_F can be negative; this only increases that coefficient.

Let P'=\eta^*\nu^*P. On a higher projective base \mu:\widehat V'\to V' where the moduli divisor is nef, its trace satisfies M_{V'}=\mu_*M_{\widehat V'}. The projection formula gives

Only this intersection is asserted to be nonnegative; the trace M_{V'} need not itself be nef. Likewise B_{V'}\cdot(P')^{f-1}\geq0 by effectivity and nefness. All intersections with Weil divisors here mean their cycle intersections with rational Cartier powers, so they are defined even when the separate canonical or discriminant divisor is not \Q-Cartier.

Choose compatible canonical Weil divisors. Since \eta is finite and separable, the codimension-one ramification formula, followed by pushforward, is

This identity of Weil cycles uses the discrete valuation rings at generic prime divisors and does not require K_{V^\nu} to be \Q-Cartier. Intersect Equation (25) with (P')^{f-1} and discard the nonnegative discriminant and moduli intersections. Pushforward through \eta and the nonnegative ramification intersection then give

Finally the pushforward of K_{V^*} to V^\nu is its compatible canonical cycle. All additional divisors on the resolution have image of codimension at least two, so their intersection with f-1 pulled-back Cartier powers is zero. Dividing by \deg\eta proves Equation (22). When f=1 there are no birational exceptional divisors on the normal curve, and the same calculation applies directly.□

Proof. The local ring R=\cO_{Z,\eta_V} is regular of dimension a. Isolation means that \mathfrak bR is \mathfrak m-primary. Choose a general sections of \mathcal H. Their local equations form a system of parameters in R, since an \mathfrak m-primary ideal is contained in no smaller prime. Let \mathfrak q be the parameter ideal they generate. It is contained in \mathfrak m^h. Regularity and the multiplicity comparison give

This is the generic intersection multiplicity along V.

The global intersection can be performed without assuming that the series has no other base components. Before each cut, discard the components of the current effective cycle that lie in the base locus. Before the final cut their dimensions are greater than f, so none can contain V: otherwise V would not be a base-locus component. Discarding them therefore leaves the calculation at \eta_V unchanged. A general next section meets the remaining components properly. Nefness of P shows that discarding effective components can only decrease their total P-degree. After a cuts the surviving effective cycle has P-degree at most k^aP^d, and contains V with multiplicity at least h^a. This proves Equation (26).

Let \lambda=\operatorname{lct}_{\eta_V}(\mathfrak b) be the generic ideal threshold. It is positive and rational. The valuation from blowing up the smooth generic center has log discrepancy a and ideal order at least h, so

Because \mathfrak bR is primary, a divisor computing this threshold has center V. We realize the ideal threshold by a divisor rather than apply subadjunction to an ideal formally. Take a log resolution of Z and \mathfrak b, so that the pulled-back system has fixed divisor F and a basepoint-free moving part. Choose sufficiently many general members D_1,\ldots,D_N of \mathcal H and put

On the resolution the fixed contribution is \lambda F. The moving contributions have coefficients \lambda/N<1 and, by general choice, meet the fixed resolution divisor with simple normal crossings on the open in question. Thus (Z,\Theta) is lc at the generic point of V and has the same lc place there as the ideal threshold. Global log canonicity is unnecessary. For f>0, Lemma 7.1 gives Equation (27), and c=k\lambda\leq ak/h.□

Proof. On the smooth marked-point open, evaluation into the finite jet bundle is a morphism from the constant bundle H^0(Z,kP)\otimes\cO to a locally free sheaf. Remove the rank-jump loci for the finitely many levels. Its kernels are then vector subbundles whose fibers are exactly the full spaces \mathcal H_i(x). Their base loci are nested in increasing order as i increases. Starting with the stipulated component through the mark, follow a component containing it at each later level. Every such component has dimension between 1 and d-1: the kernel is nonzero, so its base locus is proper. Among the d+1 nested components two successive ones have the same dimension and hence are equal.

This tracking can be made algebraic for the fixed k. Follow the finitely many components of the geometric generic base loci, their inclusions, and their incidences with the mark. They are defined after a finite extension of the generic parameter field. Spread over a corresponding irreducible parameter variety and shrink for the required flatness, geometric integrality, and constant-rank conditions. There are only finitely many choices of adjacent step; one of them gives a family dominant over the marked-point open. It gives the asserted T and \Gamma. The evaluation \Gamma\to Z is dominant because the moving mark belongs to each V_t and its map to Z is dominant.

We now prove ordinary high-order vanishing. Work on a smooth parameter open, and denote the higher-kernel bundle there by \mathcal K. Evaluation on T\times Z gives a map from the pullback of \mathcal K to the pullback of \cO_Z(kP). Let \mathfrak b_T be its base ideal. After trivializing the target line bundle, a local frame of \mathcal K generates \mathfrak b_T. We show that every frame section lies in \mathcal I_\Gamma^h. Each frame element has the form

where the s_\alpha are a fixed basis of the global space H^0(Z,kP). Parameter differentiation with z fixed only differentiates the coefficients. Every derivative, at each fixed parameter, is consequently a global section of the same line bundle kP.

Write x(t) for the moving mark. In local coordinates z-x(t), the section starts in order \lceil k\tau_{i+1}\rceil. Each parameter derivative lowers this order by at most one. A derivative of order r therefore belongs to the lower kernel if

The right side is at least k\delta-1. Since k\delta\geq4, every derivative of order less than h=\lceil k\delta/2\rceil is allowed. It thus vanishes along \Gamma, which lies in the lower base locus. A change of higher-kernel frame contributes only parameter coefficients and derivatives of orders no larger than r, so the conclusion holds for any local frame.

Here a first-order normal-space assertion must be upgraded to all orders. At a general smooth point of \Gamma, its projection to Z is submersive. Thus

In particular, parameter directions span the normal space to \Gamma. Choose parameter coordinates t=(t',t''), where t' has length a=\codim_{T\times Z}\Gamma, such that the normal equations have an invertible Jacobian minor in the t' directions. The analytic implicit-function theorem writes

Set u=t'-F(t'',z). With t'' and z held fixed, differentiation in t' is exactly differentiation in u, to every order. Use a frame of kP depending only on z. The vanishing on \Gamma of all parameter derivatives of order below h says that every normal Taylor coefficient of degree below h vanishes. Therefore s\in\mathcal I_\Gamma^h in the ordinary analytic local ring. Equivalently, in arbitrary relative normal coordinates a product of r normal vector fields expands into parameter derivatives of orders at most r, so no unaccounted higher-coordinate terms arise.

The sections and ideals are algebraic. Faithful flatness of the analytic local ring over the algebraic local ring gives the same ideal membership algebraically. Applying this to a finite frame and clearing the finitely many denominators gives containment on a dense algebraic open of the incidence. Generic smoothness of \Gamma\to T identifies its scheme fiber there with the reduced general component V_t. Restriction therefore gives \mathcal I_\Gamma^h\cO_{\{t\}\times Z}=\mathcal I_{V_t}^h locally. The specialized frame is a basis of the entire higher kernel, so the containment concerns its whole base ideal.

If Z is \Q-factorial and klt, apply Lemma 7.2 to the higher kernel. Its generic base ideal is primary because V_t is a base-locus component, and its generic point is smooth by incidence dominance. The bounds follow from k/h\le2/\delta. On the isomorphism open of the big semiample contraction the image of V_t has dimension f, so the pullback of an ample class has positive top intersection on V_t. This proves the stated positivity qualification.□

Proof. Choose the incidence point after removing the other base components, the locus where the ideal containment is unavailable, and the failures of generic smoothness for the incidence over its parameter space and for the component over its actual image. These are proper closed subsets after shrinking. A prescribed very general condition on the evaluation can also be imposed because the incidence dominates Z. The component is then chosen through this point, rather than by selecting a possibly special fiber in advance.

Locally the original base support is exactly V_t. Smoothness over the actual image makes its scheme fiber reduced at the chosen point. Thus in the ambient fiber

near that point. Restricting the actual sections extends their base ideal, which proves both asserted local properties. No other base component contains the chosen point, so none can contain the component F through it. If this component is proper in the chosen integral ambient component, the base support is proper there, and the restricted series cannot vanish identically. At its generic point the resulting ideal is primary. This is a statement about this reduced general fiber and does not assert transversality or nonzero restriction for arbitrary special fibers.□

Proof. The relative determinant is nonzero on a dense open because the general maps are generically finite in characteristic zero. Its zero divisor has finitely many irreducible components. Shrink away vertical components and make a finite parameter extension to define the geometric generic components individually. Their closures give the divisors \mathcal R_j in the statement. Shrink further to keep their fiber-image dimensions constant. These components account for every branch prime of the general finite Stein cover: a ramification prime there has a strict transform on V_t, because a proper birational morphism to a normal variety is an isomorphism at each generic divisorial point. A divisor exceptional over the Stein cover has image of codimension at least two on that cover and hence on Y, so it cannot hide a branch prime. Components confined to special parameters disappear upon shrinking.

Put

Each closed set in this finite union is proper by hypothesis, so B\ne Y. The finite normal covers are unramified in codimension one over the smooth open Y^\circ=Y\setminus B. Purity of the branch locus makes them finite étale there [32, Exposé X, Theorem 3.1]. The topological fundamental group of Y^\circ is finitely generated. Indeed, triangulate the compact semialgebraic space Y compatibly with its closed subset B [15, Theorem 1.10 in the author’s notes]. After barycentric subdivision the subcomplex for B is full. On its complement, normalize the sum of the barycentric coordinates at vertices outside that subcomplex; decreasing the other coordinates to zero gives a deformation retraction onto a finite subcomplex. Its fundamental group is finitely generated. A finitely generated group has only finitely many permutation representations in each symmetric group of degree at most e. Riemann existence [32, Exposé XII, Theorem 5.1] therefore gives finitely many finite étale covers of these degrees. A finite normal cover of Y is the normalization in the function field of its restriction to Y^\circ, and is determined by that restriction. Taking the union of the proper image closures for finitely many projections proves the simultaneous assertion.□

Proof. First, pseudo-effectivity of K gives the volume bounds:

Suppose the asserted order bound fails for arbitrarily small t. There are countably many Cartier degrees, and nonzero jet kernels in each degree are detected by algebraic rank loci. Thus failure at a very general point gives a degree in which a violating section exists on a common constant-rank marked-point open. Taking powers makes this degree k arbitrarily large and divisible. Choose n+1 levels strictly between \rho and 4\rho, with equal gaps comparable to \rho. The full jet kernels at these levels are nonzero. Write \tau_0 for the lowest level. If the marked point were isolated in its base locus, local Bezout for n general kernel sections would contribute at least (k\tau_0)^n>k^nP^n, exceeding the total intersection. Lemma 7.3 therefore supplies a moving proper positive-dimensional component V.

We can now apply the curve construction. The component is of general type by Proposition 6.3. Here let Z be the scaling model for K+2tA, so that P=2r(K_Z+2tA_Z). The effective transform A_Z does not contain a general such component, and K_Z|_V\le(2r)^{-1}P|_V in effective order. Equations (33) consequently imply Equation (34) with constants uniform in t; the gap is bounded above and away from zero because \rho is comparable to the fixed \epsilon. The resulting bounded-genus, bounded-normalized-degree covering curves contradict Proposition 6.8. The argument applies to every Cartier multiple, not only the divisible degrees used for tracking: a section violating the bound in any Cartier degree could be raised to a power in an arbitrarily divisible degree. Its order and its degree increase by the same factor. Finally, exceptional additions preserve the section spaces and the orders at general points. This proves the assertion on resolutions.□

Proof. We construct the semiample models as models of big klt adjoints. The weight decomposition then gives both the volume formulas and the comparison with the axis models.

The product Y\times Y is \Q-factorial. To see this, take a smooth resolution \widetilde Y\to Y. Its irregularity is zero by Lemma 6.1, so every line bundle on \widetilde Y\times\widetilde Y is a tensor product of line bundles from the two factors. The strict transform of a Weil divisor on Y\times Y is therefore linearly equivalent to a sum of two factorwise pullbacks. Push this relation down to Y\times Y. The resulting factor divisors are \Q-Cartier because Y is \Q-factorial, and hence so is the original divisor. Products of canonical singularities are canonical, and the projective bundles are klt. The two disjoint Cartier axes form a plt boundary D, and

The notation K_{\rm sum} means K_1+K_2, or K on a slice.

For 0<\sigma\le1, put

Both endpoint weight systems are big, so choose an effective rational representative of \Xi_t containing neither axis, and denote the representative by \Xi_t as well. To make the adjoint construction uniform, we first obtain a threshold bound independent of \sigma. Restrict to 0<t\le\sigma/(1+m). Since \gamma\ge2\sigma, the coefficient

lies in (0,1]. Thus the numerical classes \xi+aA_{\rm sum}, and their restrictions to either fixed axis, range over bounded segments independent of \sigma,q,r,t. On one fixed smooth resolution the effective pullbacks of these chosen divisors have uniformly bounded degree against a fixed very ample class. That degree bounds their multiplicity at every point, including the coefficients of exceptional components. Rational denominators do not enter this estimate. The smooth lc threshold is at least the reciprocal of maximal multiplicity.

To transfer this estimate to the fixed klt space, write the crepant boundary on its resolution as an SNC divisor with coefficients below one. The minimum of one and the positive numbers one minus its positive coefficients bounds its log discrepancies below by a fixed positive multiple of smooth log discrepancies. Therefore the preceding smooth bound gives a common klt threshold \eta>0 for \Xi_t on the original space. The same reasoning on each axis gives, after decreasing \eta, a klt threshold bound for the restricted divisor there. The first bound will be used away from the axes; the restricted bounds allow inversion of adjunction along the coefficient-one axes.

Now choose rational 0<\sigma<\min\{1,\eta/4\}, independently of q,t, and then take t small enough that t\le\sigma/(1+m) and q(1+\sigma m)/r<\sigma. It follows that \gamma<3\sigma<\eta. Inversion of adjunction with the disjoint coefficient-one axes gives plt near them; away from them the threshold bound gives klt. Lowering their coefficients to 1-\sigma consequently shows that

is klt. This choice respects the order: first \epsilon, then q, then sufficiently small t with r as in Equation (30). A direct calculation gives its adjoint class

The big klt adjoint has a good minimal model by [6]. Pushing the boundary to this model gives the stated \Delta and c_t=(1+\sigma m)/r.

We turn to the volume formulas. The weight-l summand in degree k has base classes kL_{l/k} and kL_{q-l/k}; on a slice there is one such factor. The section decomposition and asymptotic Riemann–Roch give Equations (36)–(37) as Riemann sums. One can justify passage to the integral without assuming uniform asymptotic Riemann–Roch: partition [0,q] into rational bins, compare weight classes in each bin by adding and subtracting the bin width times a fixed very ample divisor dominating both mK and -mK, take divisible-degree limits, and then shrink the bins. Volume continuity gives the formulas. Equation (31) supplies their uniform bounds and, in particular, proves bigness used above. In the range of Equation (32), the bounds needed when choosing q are explicitly

Their coefficients are independent of q; the smallness threshold for t is allowed to depend on q.

It remains to establish the effective weight comparisons. On a common resolution and in compatible divisible degrees, let F_{\mathrm{full}} and F_b be the fixed divisors of the full system and its weight-b subsystem, normalized by the degree. Since the second is a subsystem, F_b\ge F_{\mathrm{full}}. Its fixed divisor consists of the toric monomial and the base-model fixed differences, and its moving class is the sum of the indicated pullbacks of N_b. Hence the full moving class is that sum plus E_b=F_b-F_{\mathrm{full}}\ge0. This proves Equation (38). The same comparison in individual sufficiently divisible degrees shows that every section at that weight, after the full fixed subtraction, contains the corresponding multiple of E_b.

At a generic base-divisor point the small base-model maps are isomorphisms. The two endpoint systems are free there in divisible degree, so together they have no common zero on the projective-line fiber. The full system therefore has no fixed subtraction there. The same holds on a general fiber. Finally, when restricting Equation (38) to a component sweeping the torus open, choose a general component not contained in \Supp E_b. There are only countably many rational weights, so these effective restrictions can be required simultaneously.□

Proof. The tracking lemma reduces failure of the assertion to a moving base component. We will first analyze its fibers over the two base factors, and then treat components that are generically finite over both.

Write d=2n+1. Choose a gap \delta>0 large enough that

This inequality will exclude a component that is a whole projective-line fiber of a slice. Now choose q sufficiently large that

For all sufficiently small t, the d+1 levels \tau_i=2n+3+(i+1)\delta, 0\le i\le d, then lie below the volume root of the total positive part. Both choices precede the choice of t. Assume that the Seshadri assertion fails for arbitrarily small t. Because the highest level is below the volume root, jet counts show that the full kernels at all these levels are nonzero in sufficiently large divisible degree. A curve through the marked point with degree-to-multiplicity ratio below the lowest level is contained in that system’s base locus. Such a curve exists from the assumed Seshadri bound and the choice \tau_0>2n+3, whether or not the infimum defining the constant is attained. Thus the lowest kernel has a positive-dimensional base component, and Lemma 7.3 supplies a moving proper component V and both estimates (33).

The numerical estimates already rule out components of general type. For all moving components under consideration, the effective boundary \Delta of Lemma 8.2 does not contain the component. Thus K_Z|_V\le c_tP|_V in effective order. Whenever V is of general type, the curve construction following Equation (34) and weight domination give a contradiction. To treat the remaining components, we begin with their fibers over a base factor.

Restriction to a slice. Resolve the rational projection from the total model to the first copy of Y, and work on the open where this resolution and the original bundle agree. Fixing the first coordinate there gives the opposite slice. Write Z_{\mathrm{sl}} for its model from Lemma 8.2 and P_{\mathrm{sl}} for its positive part; their dimension and volume are n+1 and Equation (37), respectively. We use these slice objects until returning to the total model below. Choose one sufficiently divisible Cartier degree for the total model, the slice model, and their comparison on a common resolution. Choose a general incidence point on the common isomorphic torus open, away from the other base components and where the ordinary-power containment of Lemma 7.3 holds. Generic smoothness allows us also to require that V be smooth over its actual image in the first slot there. Let F be the reduced component of the scheme fiber through that point, and suppose 0<\dim F<n+1. Lemma 7.4 now shows that, near that point, the restricted higher-series base ideal has support exactly F and lies in the ordinary power \mathcal I_F^h, with h=\lceil k\delta/2\rceil. Because F is proper in the slice, the restricted series is nonzero and its generic base ideal is primary.

The fixed differences of the full and slice models are units on a further common open. After removing these fixed divisors and trivializing the fixed-slot lines, all restricted higher-kernel sections form a subspace of H^0(Z_{\mathrm{sl}},kP_{\mathrm{sl}}), with the same local base ideal. This subseries need not be the full jet kernel on the slice: the numerical estimates come from Lemma 7.2, applied to the actual restricted subseries. The model Z_{\mathrm{sl}} is projective, \Q-factorial and klt, and P_{\mathrm{sl}} is nef, big and semiample. Choose the incidence point also in its smooth open. The component F is proper, its generic base ideal is primary, and the preceding restriction gives ordinary multiplicity at least h\ge k\delta/2. Writing f=\dim F, the lemma therefore gives

The restricted components must also move sufficiently to apply curve exclusion. These F’s may be chosen in families sweeping the opposite slice. Indeed the incidence of the V-family dominates the total torus. Choose a general incidence point, not necessarily the original marked point, and then a general fiber of its first slot evaluation. Dominance gives the required dominant evaluation onto that fixed opposite slice. Choose a component with dominant evaluation there; generic flatness permits the fiber-component choices just made. For each fixed q,t, very general choices also avoid the exceptional sets for all rational weights. Choose this point also in the isomorphism open of the slice’s big semiample contraction and outside the relevant fixed differences. The restriction of its positive part to F is then nef and big, so its top intersection on F is positive, as required for the curve construction.

Proper base images on a slice. If F is generically finite over a proper positive-dimensional image W in the remaining base, then W, and hence F, is of general type by Proposition 6.3. The slice estimates and subadjunction give the required bounds on this component, while weight domination bounds the degrees of its projected curves. We therefore obtain forbidden bounded curves.

Over a proper base image, the other possibility is that F is saturated over its base image W: on the original bundle it is the full projective-line bundle over W. Put w=\dim W=\dim F-1. The nef weight comparison gives

Indeed P_{\mathrm{sl}}-\pi^*N_b is effective on the resolved graph of the slice projection \pi to the base, so successive mixed nef intersections give P_{\mathrm{sl}}^{w+1}.F\ge P_{\mathrm{sl}}(\pi^*N_b)^w.F. The latter is qN_b^w.W, since endpoint freeness gives fiber degree q. If W is a point, the lower bound q contradicts the slice Bezout upper bound 2(n+1)q\epsilon^n(2/\delta)^n, because our choice of \delta makes the latter strictly less than q.

Suppose w>0. The ruled component F itself need not be of general type, so we seek the adjunction estimate on its base W. We obtain it from the coefficients of the slice sections. Use the section-space identification to view the subseries on the original slice, where the toric weight decomposition is defined. The full fixed factors are units at the chosen generic torus point, so this identification preserves the order h. In degree k, expand each of these sections into its toric weights. At the generic point of W, let I_l be the ideal generated by the coefficients of the nonzero weight l over all these sections. Put h=\lceil k\delta/2\rceil. All I_l lie in \mathfrak m_W^h: vanishing along the generic torus fiber says that the coefficient of every Laurent monomial vanishes to this order. Their sum is \mathfrak m_W-primary. Otherwise its larger common zero germ, times the generic torus fiber, would contradict isolation of F.

Our aim is to construct an effective boundary \Theta\sim_{\Q}c'L_b, for some rational weight b, such that W is an lc center near its generic point and c'\le2(n-w)/\delta. Numerical subadjunction will then give a canonical-intersection bound on W, even though F is ruled. Work in the regular local ring at the generic point of W, of dimension c=n-w. Resolve the finitely many coefficient ideals simultaneously. The exponents for which their product defines an lc ideal pair form the rational polytope

Here E runs through the divisors needed to test log canonicity on the simultaneous resolution, and a(E;Y,0) is the log discrepancy. Include the exceptional divisor of the blowup of the closed point. Since every I_l\subset\mathfrak m_W^h, it gives \sum_lu_l\le c/h. The polytope is compact, and \sum_lu_l has a positive rational maximum on it. Fix a rational maximizing point. For each coordinate l, some equality at this point must involve a valuation with \ord_E(I_l)>0; otherwise u_l could be increased while staying in the polytope. Such a valuation has log discrepancy zero for the ideal pair. Its center C_l is contained in V(I_l) and contains the generic point of W. Their intersection is contained in V(\sum_l I_l), so it is exactly W locally. To apply the intersection property for lc centers, we first realize these ideal lc places by an actual divisor. Choose an integer M_0>\max_l u_l and M_0 general coefficient divisors D_{l,j} from each system. Set \Theta=\sum_l(u_l/M_0)\sum_{j=1}^{M_0}D_{l,j}. On the simultaneous resolution its fixed crepant coefficients are at most one, and its general moving divisors meet transversely with coefficients below one. Thus the pair is lc near the generic point of W, and every active valuation remains an lc place. Each C_l is consequently an lc center of this pair. The intersection property [43, Theorem 1.7], applied to the identity morphism on its lc open, shows that W is an lc center generically. This also applies when some maximizing coordinate u_l is zero: the active valuation blocking that coordinate still has center in V(I_l).

Set

Since a weight-l coefficient divisor has class kL_{l/k}, the constructed boundary has class c'L_b. Also c'\le kc/h\le2(n-w)/\delta. Transform it and W to the small scaling model Y(b); our general incidence choice places the generic point of W in their common isomorphism open. On this model the boundary has class c'N_b. Numerical subadjunction and N_b=(r+bm)K_{Y(b)}+rtA_b give

Indeed a general W is not contained in a fixed effective representative of A_b, so its contribution to this intersection is nonnegative. The slice degree estimate and Equation (39) give, in turn,

These are precisely the two bounds required in Equation (34), with constants independent of t. The variety W is of general type by Proposition 6.3; normalized curve exclusion again gives a contradiction.

A slice multisection. Proper base images have now been excluded. The remaining proper positive-dimensional slice case is \dim F=n, with F generically finite of degree e over the whole base. Weight comparison and the slice degree bound give

so e is bounded uniformly in t. Here the two axes provide the boundary needed to use log general type on the base. The closure of F on the original bundle is neither axis. Intersect this closure with each Cartier axis and push the resulting effective integral cycle to Y; call the resulting Weil divisors D_1,D_2. The two axis classes differ by the pullback of mK, and F has generic degree e over Y. The projection formula therefore gives

At the appropriate endpoint weight b\in\{0,q\}, the difference E_b|_F contains the corresponding toric-axis intersection with coefficient q. There is no full fixed subtraction above generic base-divisor points by Lemma 8.2. Intersecting with N_b^{n-1} and using mixed nef comparison yields

The other divisor has bounded degree for the same N_b, because of Equation (40) and

We now apply the logarithmic curve construction on the base Y(b), not on the multisection F. The signed divisor \pm(D_1-D_2)/(em) represents K_{Y(b)} by Equation (40). On a log resolution of this base, add the reduced strict support of D_1+D_2 and all exceptional divisors to the canonical divisor. Proposition 6.10 gives a big adjoint for the reduced strict support of D_1-D_2 and all exceptional divisors. Adding the remaining effective reduced boundary shows that the resulting log canonical divisor is big as well. Its intersection with the pulled-back N_b^{n-1} is bounded: exceptionals vanish under projection, and reduced support has degree no larger than D_1+D_2. Log effective birationality and the curve construction therefore contradict normalized curve exclusion on the base itself.

Reduction to a correspondence. We now return to the total model Z and its positive part P. The slice analysis has excluded fiber dimensions strictly between zero and n+1 over either slot. A full (n+1)-dimensional fiber over one slot would mean that V contains the whole opposite slice over its first image W. Its fiber dimension over the other slot would then be \dim W+1, strictly between zero and n+1, unless V were the full total space. That is impossible. Hence V projects generically finitely in both slots and \dim V\le n. If either image is proper, general type and the total-space estimates give the previous curve contradiction. The same is true if V is of general type. Thus only the case \dim V=n, dominant with bounded degree over both copies of Y, remains. We next show that its branch divisors cannot move; this will reduce the family to finitely many covers.

Moving branch divisors. Consider the dominating algebraic family of these correspondences, resolving maps after shrinking the parameter space. Suppose a divisorial branch component of one projection moves. On a resolution V^*, choose a ramified divisor R above its generic point. There is a rational weight b for which the effective difference E_b|_{V^*} does not contain R. Indeed take a fixed free divisible degree of P; some section does not vanish generically on R. Monomial-weight sections span that degree, so one of them does not vanish there. Only finitely many weights are tested in this fixed family; their model comparisons can be resolved simultaneously.

Let J be the branch divisor on the corresponding small axis model. Restrict the effective difference to R; this is permitted because R is not in its support. Mixed nef intersections and projection to J bound its N_b-degree by P^{n-1}.R. The ramification formula over the terminal target bounds the latter by K_{V^*}P^{n-1}: its other ramification terms are nonnegative, and the pullback of the target canonical class is pseudo-effective. The total-space subadjunction estimate now gives

The ramification coefficient of R is a positive integer, so is at least one; the other ramification and exceptional terms are nonnegative. The last bound is the total-space subadjunction estimate.

To turn this degree bound into bounded curves, we still need a canonical intersection bound on J. The required adjunction does not assume that (Y(b),J) is globally lc. Terminal Y(b) is smooth in codimension two. Thus normalization and conductor adjunction along J give

in codimension one. On resolving J^\nu, exceptional divisors map to codimension at least two and pair trivially with the pulled-back N_b^{n-2}. Consequently, with d_J=N_b^{n-1}.J,

For the second line, a general moving J is not a component of a fixed effective representative of A_b, giving the first term; mixed Hodge index gives the second. The lower bound for N_b^n and Equation (41) give the final uniform constant.

A moving J sweeps very general base points and is of general type by Proposition 6.3. Equations (41)–(42) therefore give forbidden bounded curves. Branch components can be named after a finite parameter extension and shrinking. It follows that, for all sufficiently small t, all divisorial branch components in the chosen family are fixed.

Fixed covers. We now fix t. The branch complement is chosen from the one finite-type family of correspondences under consideration, before any cover is classified. After a finite parameter extension and shrinking, resolve the two evaluation maps simultaneously over a smooth parameter open and follow the finitely many irreducible relative ramification divisors. The argument just given excludes each component whose divisorial images sweep the base: such a component would supply the bounded-genus covering curves of Proposition 6.8. For every remaining component the closure of its divisorial evaluation image is a fixed proper closed subset of Y, and the intersection estimates already bound the degrees of both projections. These are precisely the hypotheses of Lemma 7.5 for the smooth projective family of resolved correspondences. It gives a smooth dense open Y^0 and only finitely many finite normal Stein covers in either slot, all etale over Y^0. Bad parameter fibers are removed by shrinking the parameter space, not by deleting their possibly dominant images in Y. Both Y^0 and the finite cover lists may depend on this fixed t; no common complement or list as t\to0 is used.

For a general member V_a^* of the resolved family, factor the two projections through their finite normal Stein covers. The preceding finiteness lets us fix these as T_1\to Y and T_2\to Y after restricting to a family whose images still dominate Y\times Y. Each map \beta_{i,a}:V_a^*\to T_i is birational, so the two projections determine a birational map

The graphs of these maps form an algebraic family after choosing a Hilbert-scheme component and shrinking for flatness and birationality of both projections. Such a component with dominant evaluation exists: there are only countably many graph Hilbert schemes and finitely many cover choices, whereas the original incidence dominates the base product. Its graph incidence has image of dimension 2n in T_1\times T_2, because this product is finite over Y\times Y. The product is integral of dimension 2n, so the graph incidence dominates it. Proposition 6.11 excludes exactly this family of birational maps between the two fixed covers.

Every possible moving component V has now led to a contradiction, so the Seshadri bound holds. To obtain the assertion about jets, observe that at a very general point the big semiample contraction is an isomorphism onto its image near that point. The jet interpretation of the Seshadri constant for the ample class downstairs gives the stated divisible jet-separating multiple. Complete systems agree under the model comparisons, so this conclusion transfers to the original torus open and to common resolutions.□

Proof. Multiply N_p copies of the fixed jet system. Every monomial of total degree at most (2n+2)k_0N_p is a product of N_p monomials of degree at most (2n+2)k_0. Taking sections with those leading monomials gives a triangular system, ordered by total degree. Starting at degree zero and removing the error in each successive degree proves the required jet surjection. This argument takes place in the local ring and never divides by factorials.

Put d_p=(p-k_0N_p)q, so 0\le d_p\le(k_0-1)q. Among the sections fixed before reduction, choose

nonzero at the first and second base coordinates of z_*, respectively. Their tensor product belongs to the summand of first-slot weight zero in the projective-bundle formula; its fiber monomial is the d_p-th power of the second coordinate. It defines a section of d_p\xi+T_{0,1}+T_{0,2} nonzero at z_*, since z_* lies off both axes. Multiplying the product jet system by this section changes its divisor from N_pk_0M to pq\xi+B_{p,1}+B_{p,2}. Multiplication is invertible on the local jet algebra. Only the finitely many previously fixed values of d_p occur.

Since \dim Z=2n+1, the quotient by the p-th powers of regular parameters has top total degree (2n+1)(p-1). For large p this is at most (2n+2)k_0N_p. The underlying ordinary/Frobenius ideal comparison is also recorded in [52, proof of Proposition 2.12, Equation (2.8)].□

Proof. Trivialize the two summands defining Z near the selected base points. Let z be the torus coordinate, whose value at z_* is z_0\ne0. By the projective-bundle formula, the sections in Lemma 9.2 are weight polynomials in z with weight-j coefficients in

Let A_* be the tensor product of the local algebras of the two base Frobenius fibers. The local quotient in Lemma 9.2 is

free over A_* with basis 1,z,\ldots,z^{p-1}. Project to the coefficient of 1 in this basis. Exactly the weights j=pa survive, with multipliers z_0^{pa}. Their coefficients are the values of the corresponding summands of Equation (46), in the chosen frames and the induced twisted frames. As the full jet map is surjective, these coefficients span A_*, of dimension p^{2n}=R. Thus the evaluation has full rank at this pair of base points, and hence generically. The calculation uses a basis over the possibly nonreduced algebra A_*, so reducedness of the Frobenius fibers is unnecessary.□

Proof. Write F_C for absolute Frobenius of C. Langer’s instability estimate [45, Corollary 2.5] gives

To check the uniform constant, the discrepancy from \mu_{\min}(V)\ge0 is at most (\rk V-1)\deg A_C/(p-1), where A_C is a nef line bundle such that T_C\otimes A_C is globally generated. Its degree can be bounded in terms of the fixed genus, using a sufficiently positive line bundle. Here the rank is n, irrespective of the fact that the base is a curve.

We also need, for every integer i\ge0,

Fix p and i. Choose a line bundle A_e of degree -\mu_{\min}(F_C^{e*}V)+O(1), rounded up with a fixed genus-dependent additive constant, so that F_C^{e*}V\otimes A_e is globally generated. This follows from Serre duality: after subtracting any point, its minimum slope can be made greater than 2g(C)-2, which annihilates H^1 and makes evaluation at that point surjective. For every vector-bundle quotient Q of V^{\otimes i}, the bundle F_C^{e*}Q\otimes A_e^{\otimes i} is then globally generated. Consequently

Divide by p^e and let e\to\infty to prove Equation (53). This limit is taken for each fixed p,i; there is no interchange of Frobenius-iteration and characteristic limits.

Since A_{u-j}(V) is a quotient of V^{\otimes(u-j)}, the perfect pairing Equation (51) yields

□

Proof. Write the fixed flag as

At step k, fix the preceding nonnegative integers b_1,\ldots,b_{k-1} and put

For each b_k\ge0, the divisor restriction sequence is

Iterating it for b_k=0,1,\ldots and then proceeding to the next member of the flag bounds the section space by

At each stage the remainder is zero after sufficiently negative ample twisting. Its stopping index need not be uniform in j,p or in the preceding twists: enlarging the nonnegative finite sums to the displayed lattice sum preserves the inequality.

A summand in Equation (55) vanishes when \sum_i b_i(h_i\cdot C)>M_p, because then its maximum slope is negative. Each nonzero summand has at most e_j(M_p+1) sections. Indeed, evaluation at \lfloor M_p\rfloor+1 distinct points is injective; the kernel has negative maximum slope after subtracting those points. By Equation (45) it follows that

The leading coefficient follows from \Lambda=\prod_i l_i. More importantly, the whole scalar polynomial on the first line is the same for every j.

Sum over the filtration in Equation (49). The ranks satisfy \sum_j e_j=p^n, so Equation (56) gives the asserted bound with R=p^{2n}. There is no extra factor for the number of graded pieces. Finally

by Equation (43), proving the strict inequality for all large p.□

Proof of Theorem 1.1 over \C. Use dimension induction in the real-boundary category of Proposition 2.5. In dimension zero the variety is a point. Suppose the good-model assertion holds in all smaller dimensions. The signed-representative argument (Theorem 3.1), the exclusions, and the jet estimates are then available with exactly that lower-dimensional hypothesis. Theorem 9.6 establishes smooth nonvanishing in the present dimension. Proposition 2.5 therefore proves the real-boundary good-model assertion in the present dimension. This is precisely the hypothesis needed to continue to the next dimension.

Apply this to the rational lc pair (X,B) of Theorem 1.1. Let f:(X_d,B_d)\to(X,B) be a crepant dlt model and let (V,B_V) be its good log minimal model. Take a common resolution a:W\to X_d, q:W\to V, and put p=f\circ a. Then

The first equality is crepancy. The second is the nef comparison in Lemma 2.2, since K_{X_d}+B_d=f^*(K_X+B) is nef. The right side is semiample because (V,B_V) is a good model, so the pullback of K_X+B is semiample. Normality gives p_*\OO_W=\OO_X. Choose a sufficiently divisible Cartier multiple m(K_X+B) whose pullback is globally generated. The projection formula identifies its sections with those of its pullback. Surjectivity of evaluation upstairs implies surjectivity downstairs: otherwise a base point downstairs would make every pulled-back section vanish on its nonempty fiber. Hence K_X+B is semiample.□

Proof. Let (X,B) be defined over such a field k. Choose a finitely generated subfield k_1\subset k over which the projective variety, the rational boundary, a Cartier multiple of its adjoint, and a log resolution are all defined. After enlarging k_1 if necessary, these data base change to the given data. Let k_0 be its algebraic closure inside k, and denote the resulting pair by (X_0,B_0). Choose an embedding k_0\hookrightarrow\C, and write (X_{\C},B_{\C}) for the complex base change.

The chosen log resolution tests log canonicity after either algebraically closed field extension: its boundary remains simple normal crossing and all discrepancy coefficients remain unchanged. The descended pair is consequently lc over both k_0 and \C.

Nefness is also invariant under such extensions. Indeed, a curve over an extension is represented by a point of a relative Hilbert scheme after its finite defining data have been spread over a finite-type parameter scheme. The degree of a fixed line bundle is constant in the resulting flat family. Specializing to a closed point over the algebraically closed smaller field preserves a negative degree, if one existed; some irreducible component of the specialized curve would then have negative degree. This contradicts nefness over that field. Conversely, curves over the smaller field remain available after extension. Apply the converse to k_0\subset k: nefness of K_X+B gives nefness of K_{X_0}+B_0. Apply the forward direction to k_0\hookrightarrow\C: its complex base change is nef as well.

The complex case supplies an integer m>0, enlarged to a multiple of the Cartier index fixed over k_0, for which the Cartier line bundle M=\OO_{X_0}(m(K_{X_0}+B_0)) becomes globally generated over \C. Proper flat base change for sections identifies

Tensoring the evaluation map for M with \C therefore gives the surjective complex evaluation map. Its coherent cokernel is zero by faithful flatness. Extending from k_0 to k preserves this surjectivity, so the same integer m gives a globally generated line bundle on the original X.□

Proof. The simultaneous induction in Section 10 starts with dimension zero and proves the full real-boundary conclusion of Proposition 2.5 at each dimension. In particular, its conclusion applies to the given pair before any rational-boundary or nefness specialization.□

Proof. Apply Theorem 11.1 to the klt pair (X,0). Choose its \Q-factorial log minimal model, as in the construction of Proposition 2.5. The klt clause of Lemma 2.2 shows that it extracts no divisor, remains klt, and has the displayed effective exceptional comparison. Its boundary is zero, being the strict transform of the zero boundary with no extracted divisors. Discrepancies strictly improve at contracted divisors, which is the K_X-negative condition; semiampleness is the good-model conclusion. For sufficiently divisible m>0, the effective q-exceptional divisor mE satisfies q_*\OO_W(mE)=\OO_V, since V is normal. Projection formula and the displayed comparison identify the pluricanonical section spaces. Their ratios agree in the common function field, proving the assertion about Iitaka dimension.□

Proof. Choose a positive integer m such that the line bundle L=\OO_Z(mA) is globally generated. Its complete system gives

The image has dimension at most \kappa(Z,A)=0, so it is a point. A linear form nonzero at that point pulls back to a nowhere-vanishing section of L. Thus L\simeq\OO_Z, an actual line-bundle isomorphism, and A\sim_{\Q}0.□

Proof. Theorem 1.1 gives semiampleness on the original normal variety. Lemma 11.3 gives a positive Cartier multiple with trivial associated line bundle. Klt pairs are lc, so the fivefold assertion is a special case.□

Proof. Write X^\nu=\coprod_i X_i^\nu for the normalization and put \Theta_i=\Theta|_{X_i^\nu}. Each (X_i^\nu,\Theta_i) is a normal projective log canonical pair with effective rational boundary. The adjoint K_{X_i^\nu}+\Theta_i is the pullback of K_X+\Delta, so it is nef. Theorem 1.1 makes each of these adjoints semiample. There are only finitely many components, hence one common positive Cartier multiple is globally generated on their disjoint union. Thus \nu^*(K_X+\Delta) is semiample. The normalization gluing theorem [26, Theorem 1.5] now gives semiampleness on X.□

Proof. Set B=\Delta-S. The pair (X,S+B) is dlt, B is an effective rational divisor, and \lfloor S+B\rfloor=S. Since a dlt pair is log canonical, Theorem 1.1 makes its nef adjoint semiample. The pair and the displayed effective representative therefore satisfy the hypotheses of [26, Proposition 5.12], which gives the stated restriction surjectivity for every Cartier multiple with m\geq2. This is the supported extension formulation in [26, Conjecture 5.8].□

Proof. In dimension zero, X is a point, and in (ii) so is S. The boundary and canonical divisor are zero, and the two algebras are k[t] and \OO_S[t], respectively, with \deg t=1. We therefore assume \dim X\geq1.

We first prove (i). If D is not pseudo-effective, a nonzero section s in degree m>0 would give

This would make D pseudo-effective. Thus all positive graded pieces vanish in this case. Since H^0(X,\OO_X)=k, we have R(X,D)=k.

Suppose now that D is pseudo-effective. The ordinary-pair conclusion of the companion minimal-model theorem [53, Corollary 1.2] gives a normal projective lc pair (Y,\Delta_Y) and a birational map \phi:X\dashrightarrow Y extracting no divisors, with \Delta_Y=\phi_*\Delta and D_Y=K_Y+\Delta_Y \Q-Cartier and nef. Log discrepancies do not decrease. Theorem 1.1 makes the rational adjoint D_Y semiample.

On a common smooth resolution p:W\to X, q:W\to Y, choose compatible canonical divisors. Then

Indeed, the coefficient of E at a prime divisor F is a(F;Y,\Delta_Y)-a(F;X,\Delta), which is nonnegative by the discrepancy comparison. Suppose F is not q-exceptional. Since \phi extracts no divisors, F is the strict transform of a divisor on X. The pushed-forward boundary gives equality of these discrepancies. This proves the asserted support.

For any normal variety Z, rational Cartier divisor G, and m\geq0, the nonzero global sections of \OO_Z(\lfloor mG\rfloor) are exactly the rational functions s satisfying

This is equivalent to the usual inequality with \lfloor mG\rfloor because the coefficients of \operatorname{div}_Z(s) are integers. If the inequality holds on X, pull back its effective \Q-Cartier left side to W and push forward by q. Equation (67) and q_*E=0 give the inequality on Y. Conversely, pull back the inequality on Y, add mE, and push forward by p. Thus, inside the common function field,

These identifications preserve multiplication, so they identify the full graded rings.

It remains to use semiampleness on Y. Choose r>0 such that rD_Y is Cartier and globally generated. Its complete linear system defines \psi:Y\to\PP^N_k with \OO_Y(rD_Y)\simeq\psi^*\OO_{\PP^N}(1). Let A=k[T_0,\ldots,T_N] act on R(Y,D_Y) by pulling back the coordinate sections, each in degree r. For 0\leq i<r, put \mathcal F_i=\psi_*\OO_Y(\lfloor iD_Y\rfloor), which is coherent on \PP^N_k because \psi is proper. Since

projection formula identifies the part in degrees congruent to i modulo r, as an A-module, with

This is a finite A-module by the standard finite-generation theorem for section modules [61, Lemma 30.14.1(5)]. There are only r residue classes, so R(Y,D_Y) is finite as an A-module and hence finitely generated as a k-algebra. This proves (i).

For (ii), set n=\dim X. In the notation of Fujino–Gongyo, Conjecture C_{\leq n-1} asks for good minimal models of projective \Q-factorial dlt pairs with effective real boundary and pseudo-effective adjoint in dimensions at most n-1. Theorem 11.1 supplies this statement. Its good-model convention is the one fixed in Section 2, with a \Q-factorial dlt model as in Proposition 2.5. Theorem 1.1 of Fujino–Gongyo [28] therefore gives their Conjecture B_n: finite generation of the log canonical ring for projective rational plt pairs of dimension n whose adjoint is big and whose boundary has irreducible round-down. Their relative reduction, Corollary 1.4, applies to the given rational lc pair and proper morphism onto S, and gives exactly (ii) without requiring X to be projective over \C. Their Theorem 1.1 also gives (i) directly when k=\C.□

Proof. For positive-dimensional X, the theorem of Boucksom–Demailly–Păun–Peternell [8, Corollary 0.3 and Theorem 2.6] says that X is non-uniruled exactly when K_X is pseudo-effective. Corollary 11.2 then gives \kappa(X)\geq0: the semiample canonical divisor on the good model has a nonzero section in a positive multiple, and the pluricanonical section spaces agree. Conversely, a nonzero pluricanonical section makes K_X pseudo-effective. This proves (i).

For (ii), rational connectedness forces all positive covariant tensor differentials to vanish, as recalled by Lazić and Peternell. For the converse, their Proposition 2.4 reduces Mumford’s criterion in dimension n to tensor nonvanishing for smooth projective varieties of positive dimension at most n whose canonical class is pseudo-effective [46, Proposition 2.4]. If Y is such a variety of dimension d, the preceding good-model argument supplies a nonzero section of mK_Y for some integer m>0. Antisymmetrization embeds \Omega_Y^d into (\Omega_Y^1)^{\otimes d} over \C; tensoring this injection m times gives the required nonzero tensor differential. This is the observation in [46, Remark 2.5], so their reduction applies in every dimension. Both assertions for a point follow from our conventions.□

Proof. Write r=\dim R(X) in (i). Resolve the MRC map to a morphism f:X'\to R(X), with X' smooth and projective. For a smooth general fiber F, the cotangent sequence is

If p>r, every graded piece of the induced exterior-power filtration on \Omega_{X'}^p|_F contains a positive exterior power of \Omega_F^1: at most r factors can come from \OO_F^{\oplus r}. Consequently, for k>0, the graded pieces of the induced filtration on \operatorname{Sym}^k(\Omega_{X'}^p)|_F are direct summands of sums of positive tensor powers of \Omega_F^1. Here we use antisymmetrization and symmetrization over \C. Rational connectedness of F makes all their sections vanish, by Corollary 13.1. A global section therefore vanishes on every sufficiently general fiber, hence on X'. Birational invariance of global covariant tensor differentials on smooth projective varieties gives the same vanishing on X. This is the upper-bound argument of [9, Section 3].

If r>0, the smooth non-uniruled base has \kappa(R(X))\geq0 by Corollary 13.1. A nonzero section of kK_{R(X)} for some k>0 pulls back to a nonzero section of \operatorname{Sym}^k(\Omega_X^r). The pullback is taken on X' and descends to X by the same birational invariance. If r=0, the term p=0 gives a nonzero section and the same equality follows. This is the equality r=r^- predicted in the remark ending [9, Section 3].

For (ii), the forward vanishing is the rational-curve argument recalled by Brunebarbe and Campana in the same section. Conversely, the stated vanishing gives H^{2,0}(X)=H^0(X,\Omega_X^2)=0. Kodaira’s projectivity criterion then makes X projective: the real degree-two cohomology is of type (1,1), so a rational class sufficiently close to a Kähler class is still Kähler, and clearing denominators gives an integral Kähler class. This is the projectivity reduction in [9, Section 3]. Part (i) now applies. Its maximum is zero because every term with p>0 vanishes, so the MRC base is a point and X is rationally connected.□

Proof. Campana’s birationally invariant fibration properties [60, Appendix A, property 4] give \omega(X)=\omega(R(X)): the general MRC fiber is rationally connected and has \omega=-\infty. Indeed any rank-one subsheaf with nonnegative Iitaka dimension would give a section of a positive tensor power, contrary to Corollary 13.1; a point has \omega=-\infty by the empty maximum. Birational invariance permits resolving the MRC map. If R(X) is a point, the displayed formula therefore follows without changing our convention \kappa(\mathrm{pt})=0.

Suppose that R=R(X) has positive dimension. It is non-uniruled, so \kappa(R)\geq0 by Corollary 13.1. To apply Campana’s abundance deduction [60, Conjecture A.2 and Remark A.3], take a resolved Iitaka fibration R'\to Z, with R' and Z smooth and projective, R' birational to R, and \dim Z=\kappa(R). Its smooth general fiber F has \kappa(F)=0. If F is positive-dimensional, Corollary 11.2 and Lemma 11.3 give a normal birational good model V_F with K_{V_F}\sim_{\Q}0. In particular its canonical divisor is numerically trivial on its smooth locus. Campana’s criterion [60, Appendix A, Examples A.1(2)] says that the existence of such a normal birational model implies \omega(F)=0; it does not require K_F itself to be trivial. Property 4 of the same appendix and birational invariance now give \omega(R)=\omega(R')\leq\dim Z=\kappa(R). If F is a point, this upper bound is simply \omega(R)\leq\dim R=\kappa(R). The reverse inequality \omega(R)\geq\kappa(R) is one of the defining cotangent inequalities recorded in the same appendix. This proves the piecewise formula. A non-uniruled X has R(X) birational to X, giving the first stated specialization.

Campana also records \omega(X)\geq\kappa^+(X)\geq\kappa(X) [60, Appendix A, property 2]. When \kappa(X)\geq0, Corollary 13.1 makes X non-uniruled, and the two outer terms are equal by the formula just proved. Thus \kappa^+(X)=\kappa(X). This is the abundance consequence recalled by Taji [60, Theorem 1.3], who attributes it to Campana [10, Proposition 3.10].□

Proof. Corollary 13.3 gives \kappa^+(X)=0. Campana’s criterion [10, Corollary 5.3], in the form recalled by Taji [60, Theorem 1.6], gives the conclusion.□

Proof. Theorem 1.1 with zero boundary on a smooth projective complex variety gives precisely Conjecture 1.2 of Claudon–Höring–Kollár [14]. Their Theorem 1.1 and Corollary 1.5 therefore give (i) and (ii), respectively.□

Proof. Since \pi is proper, \pi^{-1}(W) is compact. Local rational Cartier indices therefore have a common multiple m_0 on an open neighborhood O of \pi^{-1}(W). The image of the closed set X\setminus O is closed and disjoint from W, so there is an open neighborhood V\supset W with \pi^{-1}(V)\subset O. Thus m_0D is Cartier over V.

Fix P\in W. Following [25, the proof of Theorem 1.10, p. 30], work over a sufficiently small neighborhood of P contained in V and take a crepant dlt blow-up p:(X',\Delta')\to(X,\Delta). Put \pi'=\pi\circ p and D'=K_{X'}+\Delta'=p^*D. For every log canonical stratum S of (X',\Delta'), including X' itself, adjunction gives an effective rational boundary \Delta_S with

The strata are normal, and this adjoint is nef over \pi'(S). On each normal projective component F of an analytically sufficiently general fiber of S\to\pi'(S), restriction gives a log canonical pair (F,\Delta_F) with effective rational boundary and nef adjoint K_F+\Delta_F=(K_S+\Delta_S)|_F. Theorem 1.1 makes this adjoint semiample, so its Iitaka and numerical dimensions agree. Thus D'|_S is abundant over \pi'(S) for every stratum: this is precisely \pi'-log abundance in [25, Definitions 2.10 and 2.13].

The nef-and-log-abundant theorem [25, Theorem 1.4] now gives a relatively generated multiple of D' over a neighborhood U_P\subset V of P. Enlarge its integer to a multiple m_P of m_0; tensor powers preserve relative generation. Normality gives p_*\OO_{X'}=\OO_X, and projection formula identifies the direct images over U_P of \OO_X(m_PD) and \OO_{X'}(m_PD'). Their evaluation maps are related by pullback. Generation upstairs therefore implies generation downstairs: a base point downstairs would make every pulled-back section vanish on its nonempty fiber.

Choose finitely many of these neighborhoods U_{P_1},\ldots,U_{P_s} covering W, and let m be a common multiple of m_0,m_{P_1},\ldots,m_{P_s}. Put U=\bigcup_i U_{P_i}. The divisor mD is Cartier over U, and its line bundle is relatively generated on each U_{P_i}. Surjectivity of evaluation is local on the base, so it holds over U, as asserted.□