Integrable metrics and effectivity with controlled boundary

1. Introduction

A pseudoeffective divisor need not have a section in any positive multiple. This paper studies an analytic condition that makes such a divisor effective up to rational linear equivalence, after a correction on a prescribed boundary. The condition is integrability of the canonical divisor section for a positively curved anticanonical metric. Its usefulness is birational: a correction on the base of a fibration can disappear after pullback and pushforward to the original variety.

If the squared norm in a local frame of a singular metric h is e^{-\phi}, its multiplier ideal \mathcal J(h) consists of the holomorphic germs u with |u|^2e^{-\phi} locally integrable. Thus \mathcal J(h)\supset\mathcal O_Y(-C) means precisely that the canonical section s_C of the effective integral divisor C has locally finite squared norm. We call a rational divisor rationally effective if it is rationally linearly equivalent to an effective rational divisor. A rational divisor is pseudoeffective when its numerical class lies in the closure of the cone generated by effective divisors. On a smooth projective variety this is equivalent to the existence of a singular Hermitian metric with semipositive curvature current on its rational line bundle.

The support conclusion distinguishes the theorem from numerical nonvanishing. It permits the correction to be removed under a specified birational map. The analytic assumption also distinguishes two notions of integrability: s_C may have finite norm even when a nonvanishing local frame does not. The stronger, unweighted condition will yield a separate finite-generation result.

1.1. The boundary construction and its antecedents

An effective rational boundary \Delta is klt (Kawamata log terminal) if, on a log resolution \sigma:Y'\to Y, every coefficient of \sigma^*(K_Y+\Delta)-K_{Y'} is less than one. For a simple normal crossing boundary on a smooth variety, this amounts to each boundary coefficient being less than one.

The proof connects multiplier-ideal approximation with big-boundary nonvanishing. Nadel vanishing [25], followed by very ample regularity, makes the relevant multiplier-ideal systems globally generated. Point extension of Ohsawa–Takegoshi [26], in the weighted Bergman approximation developed by Demailly [15], compares the fixed divisor of a complete system with the metric’s singularities. On a resolution \sigma:Y'\to Y, weighted integrability produces a signed divisor \Psi=\Psi^+-\Psi^- whose coefficients are strictly below one. Its negative part lies only over C and the exceptional locus.

The approximation leaves an ample rational class A in reserve. For a prescribed pseudoeffective M, choose a small rational t>0 so that A+tM remains ample. General free and ample representatives then turn the positive part of the signed divisor into an effective big klt boundary \Delta, with

This is the pair to which the nonvanishing theorem of Birkar–Cascini–Hacon–McKernan [8, Theorem D] applies. A finite rational linear system upgrades its real-linear conclusion to rational linear equivalence. Pushforward gives the correction \sigma_*\Psi^-/t, supported on C. Sections 2 and 3 give this construction. The latter also retains an analytic perturbation argument: strong openness of Guan–Zhou [19] and small-exponent integrability of Skoda [33] permit a small pseudoeffective perturbation of the metric while preserving the integrability of the distinguished section.

A different consequence begins with an unweighted integrable semipositive metric that is smooth and strictly positive on one ordinary open set. Boucksom’s volume criterion [9] first gives bigness; a small mixture with a strict metric then gives a global curvature lower bound while preserving integrability. Section 4 makes this preparation explicit and permits perturbation in finitely many arbitrary rational directions. The resulting klt boundaries fit the multigraded adjoint finite-generation results of Cascini–Lazić [13] and Corti–Lazić [14, Theorem 3.2(1)]. More precisely, if -K_Y+A_0 has this unweighted metric, then for every finite list of rational divisors P_1,\ldots,P_s there are a common rational \epsilon>0 and Cartier multiples B_i=k_0(A_0+\epsilon P_i) whose multigraded section ring is finitely generated (Theorem 4.3). This controls the sections of specified Cartier divisors, not merely a numerical cone.

1.2. Anticanonical nonvanishing and fixed-error conversion

For a smooth projective variety X, anticanonical nonvanishing asks whether some H^0(X,-mK_X), m>0, is nonzero when -K_X is nef. Numerical effectivity is weaker: an effective divisor in the numerical class need not supply a section of the actual line bundle. For pairs (X,\Delta), the corresponding divisor is -(K_X+\Delta), the anti-log-canonical divisor. Lazić–Matsumura–Peternell–Tsakanikas–Xie [22, Theorem A] established numerical nonvanishing for projective log canonical threefold pairs with nef anti-log-canonical divisor, assuming either that the variety is \mathbb Q-factorial or that it has rational singularities. Müller [24, Corollary B] proved nonvanishing for projective klt threefold pairs with nef anti-log-canonical divisor. His higher-dimensional theorem assumes, in addition to nefness, that the anti-log-canonical divisor of the induced pair on the general fiber of the maximal rationally connected fibration is semiample [24, Theorem A].

Our principal conversion result uses smooth metric semipositivity and works in every dimension. Let X be smooth connected projective, let L=-K_X carry a smooth semipositive Hermitian metric, and let P be any fixed pseudoeffective Cartier divisor. Then Theorem 5.1 gives the implication

The fixed error P may itself have no effective representative. A single Iitaka base converts the effective divisors into a pseudoeffective rational divisor on that base. The controlled-boundary theorem makes this divisor rationally effective with correction on base primes whose pullbacks disappear over X. A rational function from the base then corrects the original rational section without introducing a pole on X.

Twisted differential forms supply the sequence through the determinant method of Lazić–Peternell [23, Lemma 4.1], adapted in [22, Lemma 5.1]. Ou’s generic tangent-nefness theorem [30, Theorem 1.4], in the cotangent-subsheaf form [22, Theorem 4.1], makes the negative determinant pseudoeffective. Our extraction argument starts with positive unbounded exponents and also includes numerically trivial L. Thus nonzero sections of (\Omega_X^1)^{\otimes p}\otimes m_jL for one fixed p\ge0 and unbounded positive m_j give an anticanonical section by Corollary 5.6. The same holds for exterior forms. This is an ordinary conversion; invariant sections require further control of characters.

1.3. Volume metrics and the distinct geometric refinements

Let X be smooth projective and let L=-K_X carry a smooth semipositive metric. In a resolved rational fibration

assume Z,Y are smooth projective, \pi is birational, and g is surjective. Write K_{Z/X}=K_Z-\pi^*K_X for the relative canonical (Jacobian) divisor. Its canonical section belongs to K_Z+\pi^*L. When \operatorname{rank}g_*\mathcal O_Z(K_{Z/X})=1, its squared fiberwise L^2 norm is a volume density \rho on the base. On the smooth base-change open where this section generates the adjoint line, -\log\rho is a metric on -K_Y. Berndtsson’s direct-image theory supplies its positivity [5, 6]. Extending it over the boundary with enough integrability is the geometric task preceding effectivity.

The main conversion proof uses a singular coefficient metric whose curvature dominates a positive base form and whose total-space multiplier ideal is trivial. Singular adjoint direct-image positivity of Păun–Takayama [31] then gives a metric with a global strict lower bound on -K_Y+C. Its integral boundary has exceptional pullback, and the distinguished section is locally square integrable.

Keeping the coefficient metric smooth gives a different output. The coarea formula and a primitive-representative curvature calculation produce an integral correction C with \pi_*g^*C=0 and

The metric is smooth and strictly positive on one ordinary open set. The higher integrability exponent permits a mixture with a globally strict metric while keeping s_C integrable. Thus the global bound needed for Theorem 1.1 follows from an explicit additional step.

The companion Metric descent and rank-preserving contractions supplies the toroidal-volume construction used here [29, Theorem 2.1]. For a normal projective model p:W\to X and a surjective equidimensional toroidal map f:W\to Y, its theorem assumes adjoint rank one at degree zero on a smooth resolution and a patch, away from the Jacobian divisor, where the pulled-back curvature dominates a positive base form. It produces an effective rational correction A_0 with

and a semipositive metric on -K_Y+A_0 with unweighted locally integrable squared frame norms, smooth and strictly positive on a nonempty ordinary open set. That companion proves the toroidal Jacobian calculation, the all-valuation integrability and the boundary extension. We use its metric after checking the stated model, rank and curvature premises in each application. The rational correction is used directly in Corollary 5.4; no rounding or replacement of its unweighted metric by an integral-boundary hypothesis is needed.

For the invariant problem, Müller [24, Theorem C] obtains naturally invariant sections for projective sub-log-canonical pairs with semiample anti-log-canonical divisor and a commutative linear algebraic group action. Theorem 9.2 here assumes X smooth connected projective, -K_X smoothly semipositive and \chi(X,\mathcal O_X)\ne0. For any algebraic torus acting on X, it gives

with the natural tangent-determinant linearization. The companion invariant-index theorem, including its actual value at zero, supplies invariant cohomology [28, Theorem 1.1]; compact-averaged hard Lefschetz then gives forms with the coefficient shift m+1. The local determinant and character arguments supply the sections needed here. The finite-cover structure and norm used in the isometric application are those of the companion headline paper, [27, Theorem 2.1 and Lemma 2.2], which develops the geometric work of Demailly–Peternell–Schneider and Campana–Demailly–Peternell [16, 11]. Our algebraic boundary criterion, smooth coarea proof and correction ring are independent of that application.

Finally, a suitable auxiliary base has an effective big klt boundary \Delta with K_Y+\Delta\sim_{\mathbb Q}C for an effective rational C supported on divisors disappearing over X. When \kappa(Y,C)=0, a log terminal model of the base is of Fano type: it admits an effective rational boundary \Theta with klt singularities such that -(K+\Theta) is ample. This is a statement about the auxiliary model. Its proof retains the precise contracted-prime support needed to return the resulting divisors to X.

The main route is the boundary and ring results followed by the single-Iitaka conversion in Section 5. Sections 6 and 7 retain the smooth coarea estimate and direct jet-separation proof of bigness; their applications reuse the common return of sections. Section 8 then develops the supported Fano-type model. The invariant base constructions in Sections 9 and 10 lead to compact-monodromy descent in Section 11. Finally, Section 12 constructs invariant tensors from extremal characters of the full equivariant Euler characteristic. That construction is independent of the weight-zero index input and descends tensors before applying ordinary conversion.

2. From an integrable section to a rational boundary

The analytic information in Theorem 1.1 concerns one section, rather than a nonvanishing local frame. The distinction already appears on a disk. For a weight \phi=a\log|z|^2 with a\ge0 and a section s_J=z^j with j\in\mathbb Z_{\ge0}, the two integrability thresholds are

On a resolution, the Jacobian contributes an additional vanishing order. After dividing the fixed divisor by the multiple used for approximation, subtracting this Jacobian order and that of J will leave coefficients strictly below one. The resulting signed divisor has a positive part that can enter a klt boundary, while its negative part records the support on which a correction is permitted.

We use additive notation for rational line bundles. If a metric has local squared frame norm e^{-\phi}, its curvature is positive when \phi is plurisubharmonic. A global strict lower bound means \sqrt{-1}\partial\bar\partial\phi\ge c\omega for one Kähler form \omega and one c>0. Smooth changes of weight do not affect local integrability. The multiplier ideal \mathcal J(\phi) consists of holomorphic germs u for which |u|^2e^{-\phi} is locally integrable. Section 4 uses the unweighted case J=0; it first obtains a global strict bound while preserving integrability of squared frame norms.

The strict inequalities are the essential output. A non-strict inequality would not produce a klt boundary. The local extension argument explains why integrability of the distinguished section is precisely the hypothesis that survives after subtracting \sigma^*J.

The signed identity supplies more information than an effectivity conclusion. In particular, its negative part has an explicit support. It does not assert integrability of the norm of a nonvanishing frame; that stronger hypothesis enters Section 4.

3. Effectivity with a prescribed correction support

The signed approximation separates two kinds of data: the negative part is confined to the permitted correction support, while the ample part can absorb a small multiple of any prescribed rational divisor. We use that ample part to insert the pseudoeffective divisor of Theorem 1.1. The resulting boundary is effective, big and klt, so big-boundary nonvanishing applies to its adjoint. We first record the passage from its real-linear conclusion to the rational linear equivalence required here.

When C=0, the correction vanishes. For nonzero C, it is the integrability of s_C that controls the support of \Psi^-; no unweighted frame-integrability hypothesis has been imposed. The divisor to which nonvanishing is applied is the effective boundary (3.1).

3.1. Preserving integrability under a pseudoeffective perturbation

One can also insert M at the level of metrics, before approximation. This gives analytic information in addition to the effectivity proved above: a small pseudoeffective perturbation preserves the integrability of the distinguished section and the global strict curvature bound.

Subtract a sufficiently small ample rational A from the perturbed metric, retaining its strict lower bound. Applying Lemma 2.1 to -K_Y+C+\epsilon M-A with r>1 then gives a second boundary construction. For its signed divisor \Psi and a general free member D_{\mathrm{mov}}, add a general small-coefficient representative U\sim_{\mathbb Q}A. Then \Delta=\Psi^++D_{\mathrm{mov}}/r+\sigma^*U is effective, big and klt, and K_{Y'}+\Delta\sim_{\mathbb Q}\epsilon\sigma^*M+\Psi^-. Nonvanishing, rational recovery and pushforward apply as before. Here the approximation itself depends on the metric on M; the support conclusion is unchanged.

4. Unweighted integrability and nearby adjoint rings

The criterion just proved accepts a weighted integral and a global strict curvature bound. For finite generation we instead begin with integrable squared frame norms and strict positivity on one ordinary open set. This permits perturbation in arbitrary finitely many rational directions. We first make the passage from local strictness to a global lower bound while preserving integrability.

4.1. Direct selection of an integrable divisor

There is a second way to perform the last step of Lemma 4.1. It chooses a section by averaging negative powers of its norm, instead of using a general member on a log resolution. This gives a direct analytic description of the effective klt representative.

Let D be a rational line bundle with a globally strictly positive metric of weights \phi_D, and suppose e^{-\phi_D} is locally integrable. For a large divisible integer m>1, take an orthonormal basis s_1,\ldots,s_N of the global sections of mD that are square integrable with weight m\phi_D and a fixed smooth volume form. The global Bergman estimate is

on each relatively compact coordinate patch. We recall the construction because this version uses global sections of mD without a fixed auxiliary twist.

Fix a point y where \phi_D(y) is finite. Point extension on a coordinate ball gives a local holomorphic section taking squared value e^{m\phi_D(y)} at y, with uniformly bounded weighted norm. Multiply it by a cutoff equal to one on a smaller ball and solve the resulting \bar\partial equation. In top holomorphic degree the coefficient line is mD-K_Y. Add a cutoff logarithmic pole d\log|z-y|^2 to the estimating weight, where d=\dim Y. For sufficiently large m, the strict curvature of mD absorbs both the fixed canonical curvature and the negative curvature of the cutoff pole, uniformly for y in a smaller ball. The correction is holomorphic near y; its integrability against |z-y|^{-2d} forces its value there to vanish. The corrected global section retains the specified value and has uniformly bounded m\phi_D-weighted norm.

For the singular-weight solution, first choose rational trivializing sections of an integral multiple of D and of K_Y. Enlarge the support of their zeros and poles by an effective divisor so that the resulting divisor is ample. Its complement is affine and trivializes the required integral line bundles. Use Stein exhaustions of that complement, approximate the psh weights on relatively compact exhaustion sets and pass to weak limits. In bidegree (d,1) the L^2 estimate uses contraction with the inverse positive line-curvature matrix; its bound is independent of an auxiliary complete Kähler metric chosen to dominate a fixed reference metric. The estimates therefore persist through the exhaustion. Since the original psh weights are locally bounded above, weighted square integrability implies ordinary square integrability near the removed divisor. The resulting holomorphic section extends across it. The local weighted estimate is Demailly’s Bergman approximation [15, Section 1, Theorem 1.2], with the point-extension input of Ohsawa–Takegoshi [26]. The strict global curvature is what permits the untwisted global construction just given. Expanding the constructed section in the orthonormal basis proves (4.1); the estimate is automatic on the polar set of the weight.

It follows that (\sum_j|s_j|^2)^{-1/m} is locally integrable. Average over the unit sphere in \mathbb C^N. Unitary invariance gives

For N\ge2, the possible singularity is an inverse power of one complex coordinate and c_{m,N}<\infty because m>1. For N=1 the identity is immediate. Fubini on a finite covering gives a nonzero section s of mD with |s|^{-2/m} locally integrable. On a log resolution, change of variables identifies this condition with all discrepancy coefficients of \operatorname{div}(s)/m being less than one. Thus \operatorname{div}(s)/m is the required effective klt rational representative of D.

5. Removing a fixed pseudoeffective error

The controlled-boundary theorem becomes useful for anticanonical nonvanishing when the boundary on an auxiliary base disappears after pullback and birational pushforward. We now construct precisely such a base. The error in the following theorem is an arbitrary fixed pseudoeffective divisor; it need not be effective.

We use the Iitaka fibration of just one N_j. Ratios of its sections force all subsequent divisors to vary affinely on the generic fiber. The vertical slopes define a pseudoeffective divisor on the base. A rank-one adjoint direct image supplies the metric required by Theorem 1.1, and its boundary is exceptional over X. Thus the correction allowed by that theorem causes no poles in the section returned to X.

5.1. Vertical minima and return of sections

We record the return step once, including the exceptional divisors that can appear when a smooth model ceases to be equidimensional.

The required model condition follows from flattening, rather than from an assertion that a final smooth resolution is flat. Starting with a resolved rational map and its Stein factorization, flatten over a modification of the base [32, Theorem 5.2.2]. Normalize the main component, resolve the base, flatten again if necessary, and then resolve the source. On the intervening flat model a prime cannot map into base codimension two: its inverse image has dimension at most \dim X-2. Any prime on the final resolution with that property is consequently exceptional over the intervening model, and hence over X.

5.2. The Iitaka base and its strict metric

5.3. The smooth toroidal metric for the same base

A toroidal embedding is locally modeled on a toric variety with its dense torus; its boundary corresponds to the toric boundary. We use strict toroidal embeddings, whose boundary components are normal. A toroidal morphism is locally a toric morphism, described on the dense tori by monomials. This local description keeps track of the exact ramification exponents in the metric correction.

There is another way to supply the metric on the Iitaka base. It keeps the original smooth anticanonical metric throughout fiber integration and gives unweighted integrability. We give the geometric verification because the two metric constructions have different conclusions.

We also isolate the algebraic consequence of unweighted integrability.

Prepare the rational Iitaka map birationally as an equidimensional strict toroidal morphism f:W\to Y, with W normal, Y smooth, and p:W\to X birational. Such a preparation follows from weak toroidalization [1, Theorem 1.1] and equidimensional toroidal subdivision [2, Proposition 4.4]. Mark the nonisomorphism locus in the source boundary before these operations. No finite base alteration or reduced-fiber assertion is used here. Take a smooth resolution Z\to W, with maps \pi:Z\to X and g:Z\to Y. The resolution itself need not be equidimensional. The generic rank-one argument (5.3) is unchanged. Likewise the horizontal interpolation and the vertical minima may be computed on W, where every vertical prime dominates a base prime.

Equation (5.1), pulled to Z, says that a multiple of [\Theta_{\pi^*L}] dominates the pullback of a positive base class pseudoeffectively. Lemma 5.3 gives a horizontal strictness patch above the smooth base-change locus, away from the Jacobian divisor. The Jacobian section is nonzero there. The model, rank-one, and strict-patch hypotheses have now been verified for the original smooth coefficient metric. These are the hypotheses of the toroidal-volume theorem in [29, Theorem 2.1]. That theorem supplies a rational A_0\ge0 with p_*f^*A_0=0 and the unweighted integrable metric of Corollary 5.4. Its coefficients are

The all-valuation integrability and extension assertions here are part of that theorem, rather than consequences of generic rank one alone. Corollary 5.4 and Lemma 5.2, on a common resolution, complete the same conversion. The output here is unweighted integrability for a rational correction: a nonvanishing frame has finite norm. The smooth coarea construction in Section 6 instead gives an integral correction and an L^{1+\eta} bound for the distinguished section. Lemma 6.5 converts that output to the same supported effectivity premise in Lemma 5.2.

5.4. The field of all pseudoeffectively dominated systems

A second base construction uses every effective system dominated by a multiple of L. It is useful when no distinguished divisor should be chosen in advance. We retain the different maximality argument and the explicit integral correction it produces.

Call a subsystem of an integral divisor D admissible if it has a nonzero section and bL-D is pseudoeffective for some integer b>0. Choose finitely many admissible systems whose joint image has maximum dimension d. Every further admissible ratio is algebraic over this image field. Take its relative algebraic closure and a smooth projective connected-fiber model \pi:Z\to X, f:Z\to Y, prepared by flattening as before. For every ample H on Y there is a b_0>0 with

Indeed the sum of the chosen divisors dominates the pullback of the product polarization; its restriction to the joint image is big. The sum itself is pseudoeffectively dominated by a multiple of L, and a large multiple of the product polarization dominates H. If d=\dim X, this makes L big.

Take sections u_j defining N_j\sim m_jL-P, put r=m_2-m_1, and let t=\pi^*(u_2/u_1), a rational section of r\pi^*L. For j>2 the expressions u_j^r u_1^{m_j-m_2} and u_2^{m_j-m_1} belong to the same admissible system, namely a multiple of m_2L-P. Writing D_t=\operatorname{div}(t) gives

Thus D_t has nonnegative horizontal part, and its divisor of vertical minima M is pseudoeffective by the same fixed-error inequality used in (5.2). The case d=0 is already settled.

For d>0, the full adjoint sheaf f_*\mathcal O_Z(E), E=K_{Z/X}, has rank one. Otherwise two generic-fiber independent sections can be made global in E+kf^*H. After pushforward their effective divisors give a subsystem of k\pi_*f^*H on X whose ratio is not in \mathbb C(Y). By (5.5) this system is admissible, contradicting maximality. Mixing a positive-current metric furnished by (5.5) with the smooth pullback metric again gives a globally defined coefficient h with \Theta_h\ge\delta f^*\omega_Y and \mathcal J(h)=\mathcal O_Z. The singular direct-image theorem therefore applies to the full actual sheaf as in the first proof. Let w be the pushed-forward Jacobian volume density on its smooth base-change locus. Then -\log w has curvature at least \delta\omega_Y there, and w has finite local mass.

Here the boundary can be written directly. Set

It is effective and integral. If a prime over Q is nonexceptional then its coefficient in E is zero, so c_Q=0. Hence f^*C is exceptional, including the primes over codimension two by the model condition. Near a general point of a prime D attaining the minimum, write f=(z_1^e,z_2,\ldots,z_d) and a=\operatorname{ord}_D E. A plurisubharmonic coefficient weight is bounded above on a relatively compact chart. Its metric norm is therefore bounded below by a positive constant times a smooth norm. Integration of this chart gives

If a=qe+s, 0\le s<e, then (a+1-e)/e=q+(s+1-e)/e\le q=\lfloor a/e\rfloor. It follows that \psi=-\log w+\log|s_C|^2 is locally bounded above at general points of every missing divisor. Extend plurisubharmonically across these divisors and then across codimension two, subtracting local potentials of \delta\omega_Y to preserve the lower bound. This gives a metric on -K_Y+C with global strict curvature and

Theorem 1.1 applies to M. Lemma 5.2, now with the rational section t, completes the alternative construction. Thus its distinct ingredients are the field of all admissible systems, full rank one from maximality, and the explicit floor correction; the algebraic return is unchanged.

5.5. From twisted tensors to a fixed error

The determinant method of Lazić–Peternell [23, Lemma 4.1], and its formulation in [22, Lemma 5.1], provides the link to twisted differentials. Those cited extraction statements include a numerical-nontriviality hypothesis. The following direct construction also covers a numerically trivial L, because the starting exponents here are positive and unbounded.

5.6. A movable-slope proof of the determinant input

For completeness we give a second proof of the pseudoeffectivity used in Lemma 5.5, under its smooth-metric application hypotheses. This proof explains the ramification correction in the relative-volume argument. It uses movable slope theory [18], duality between pseudoeffective divisors and movable curves [10], and the positive-minimal-slope algebraicity theorem of Campana–Păun [12, Theorem 1.1].

For a torsion-free sheaf \mathcal E and movable curve class \alpha, its slope is \mu_\alpha(\mathcal E)=c_1(\mathcal E)\cdot\alpha/ \operatorname{rank}\mathcal E. The symbols \mu_{\max,\alpha} and \mu_{\min,\alpha} denote the greatest and least slopes in its Harder–Narasimhan filtration, whose successive quotients are semistable with decreasing slopes.

Suppose a saturated subsheaf of a cotangent tensor power on smooth projective X had determinant of positive degree against a movable curve class \alpha. Tensor-product slope inequalities imply \mu_{\max,\alpha}(\Omega_X^1)>0, equivalently \mu_{\min,\alpha}(T_X)<0. Since c_1(T_X)\cdot\alpha\ge0, the positive-slope part \mathcal F of the Harder–Narasimhan filtration of T_X satisfies

The bracket obstruction from \bigwedge^2\mathcal F to T_X/\mathcal F vanishes by the slope inequality. Thus \mathcal F is a foliation, and the cited algebraicity theorem realizes it as the relative tangent foliation of a rational map with connected general fibers. Prepare an equidimensional model of that map, then take a smooth resolution with maps \mu:W\to X and f:W\to Y. Put L=-K_X and retain its smooth semipositive metric.

The relative adjoint bundle K_{W/Y}+\mu^*L has a nonzero section on a general fiber: there it is represented by the effective Jacobian K_{W/X}. On the smooth bundle and base-change locus, evaluation with the L^2 direct-image metric supplies its Bergman weight

in a local relative-adjoint frame. Semipositivity follows from the dual-norm form of direct-image positivity [5, 6]; zeros of the evaluation map give allowed singularities. On the isomorphism locus of \mu, the submersive differential identifies this line with \det(T_X/\mathcal F). The only extra codimension-one issue is ramification.

A prime of X vertical for the map dominates a base prime on the normal equidimensional model. Near its general point, and hence on its strict transform in W, choose coordinates

The differential contributes z_1^{e-1}. Therefore \det(T_X/\mathcal F) corresponds to the relative adjoint line twisted down by (e-1)\{z_1=0\}. A relative form written in total-space/base canonical frames acquires the factor (ez_1^{e-1})^{-1} when restricted to a fiber. Integration on a fixed local fiber chart and the holomorphic submean estimate consequently bound every unit-norm section coefficient by C|z_1|^{e-1}. It follows that

After the indicated twist, the induced weight is locally bounded above at the general point of every such prime. At horizontal primes the ordinary submean estimate gives the same upper boundedness without a ramification term. These weights extend plurisubharmonically across codimension one, and then across the remaining codimension-two locus on X. They define a semipositive singular metric on \det(T_X/\mathcal F), making that line pseudoeffective. This contradicts its negative degree against \alpha.

By movable-cone duality, the negative determinant of every cotangent tensor subsheaf is therefore pseudoeffective. The argument retains the factor e-1 because omitting it would construct a metric on the wrong line at ramified divisors.

6. Smooth coarea metrics and a dominated base

Integration along a fibration turns an anticanonical metric upstairs into an adjoint metric on its base. In this section the coefficient metric is smooth. This regularity gives a direct curvature calculation and, by coarea, an integrability exponent strictly greater than one. Together these facts produce a boundary correction supported only on base divisors whose inverse images are exceptional over the original variety.

We first establish the analytic statement, keeping curvature and integrability separate. We then construct a maximal base dominated by the anticanonical class and verify the metric hypotheses on that base. The controlled-boundary criterion converts the resulting numerical pseudoeffectivity into rational linear effectivity.

6.1. The primitive representative and horizontal strictness

For a higher-dimensional base we apply Lemma 6.1 to small embedded disks in its tangent directions. In rank one a local holomorphic frame may be chosen Chern-normal at a specified point. Dividing (6.1) by its squared norm then gives the curvature of minus the logarithm of that norm. A lower bound by the pullback of a positive base form on one patch yields strict positivity in every nonzero tangent direction at the image point. If the relative dimension is zero and the generic rank is one, the smooth map is locally an isomorphism and this assertion follows directly from the coefficient metric.

6.2. A coarea estimate

6.3. Extension across base divisors

6.4. From local strictness to a global strict metric

The metric just constructed is strictly positive on one open set. The next step uses the extra integrability exponent to introduce a global lower curvature bound while preserving the distinguished integrable section.

6.5. The smooth metric on the maximal dominated base

We now apply the smooth coarea construction to the maximal base of Section 5.4. That base already supplies the interpolation and the pseudoeffective divisor of vertical minima. The additional points here are a valuation proof of the model condition and a direct intersection argument for horizontal strictness. These allow us to use the original smooth metric, and hence retain the exponent 1+\eta in Proposition 6.3.

Retain the hypotheses of Theorem 5.1, with effective integral divisors N_j\sim a_jL-D, where D is pseudoeffective and a_j is a strictly increasing unbounded sequence of positive integers. A rational map is dominated by L if, on a smooth resolution X\xleftarrow{\ g\ }Z\xrightarrow{\ f\ }Y, one has

for an ample H on the projective base and a rational \epsilon>0. Include the point map. This is the map formulation of the admissible systems used in Section 5.4. Indeed, an admissible system gives this inequality after resolving its base ideal. Conversely, after multiplying H so that it defines a projective embedding, its pulled-back sections push down to a subsystem of |g_*f^*H|; pushing down the domination inequality makes that system admissible.

The condition persists on higher source resolutions and under birational base modifications. It also persists under a generically finite change of base to which the rational map lifts dominantly: the pullback of the old ample class is big and dominates a small ample class. Finally, the joint image of two dominated maps is dominated. On a common source resolution, add suitable positive multiples of their inequalities and use the sum of the pulled-back polarizations, whose restriction to the joint image is ample. Thus the maximality construction in Section 5.4 provides a base field \mathbb C(Y), relatively algebraically closed in \mathbb C(X), containing the functions supplied by every dominated map.

For the resulting smooth projective diagram put

The interpolation argument of Section 5.4 gives \mathbb Q-principal divisors U_j on Y with

For j>2, the two effective divisors in (a_j-a_1)N_2\sim(a_j-a_2)N_1+(a_2-a_1)N_j lie in a multiple of the admissible system N_2, so their ratio belongs to \mathbb C(Y). In particular P has nonnegative horizontal part. If Y is a point, pushing P down already proves the theorem. Assume henceforth that b=\dim Y>0.

Preparing vertical primes by their valuations

The model can be arranged so that every vertical prime not exceptional over X dominates a base prime. The flattening argument of Section 5 gives one construction. The following valuation argument gives another and explains why a base modification suffices.

On an initial model only finitely many source primes map into base codimension at least two: take an open set over which f is equidimensional and inspect the divisorial components of its complement. The valuation of such a prime restricts nontrivially to \mathbb C(Y), since a local function vanishing on its proper center has positive order. The restricted discrete valuation is divisorial. To see this, its residue transcendence degree can drop by at most \dim X-\dim Y. Residues algebraically independent over the restricted residue field lift to elements algebraically independent over \mathbb C(Y): a polynomial relation, with coefficients scaled to have minimum valuation zero, would give a residue relation. The source residue field has transcendence degree \dim X-1. The restricted residue field therefore has transcendence degree at least b-1, and the valuation inequality gives the opposite bound.

Choose a projective base model on which these finitely many valuations have divisorial centers, and resolve it. One can construct such a model by resolving maps to copies of \mathbb P^1 defined by b-1 independent residues: the center then has dimension at least b-1 and remains proper. Take a common smooth resolution of the previous source and the graph over the new base. Primes already dominating base divisors retain that property. New source primes exceptional over the old source are also exceptional over X. This proves the required preparation. The domination and field properties persist by the preceding observations. Retain P,B for their pullbacks to this model and U_j for the corresponding principal divisors on the modified base; (6.8) is unchanged.

On this prepared model define

The sum has finite support. Formula (6.8) makes W pseudoeffective by the same fixed-error limit as in Section 5.4: if c_G=\min_{J\mapsto G}\operatorname{mult}_JB/ \operatorname{mult}_Jf^*G and C=\sum_Gc_GG, then a_jW+U_j-C\ge0; dividing by a_j gives numerical classes converging to [W].

Let \mathcal S consist of base primes whose dominating source primes are all g-exceptional. It is finite, and the model condition makes the entire pullback of each member exceptional over X. Lemma 5.2, with its R=P and M=W, reduces the proof to making W+C_{\mathcal S} rationally effective for an effective correction supported on \mathcal S.

Rank one and horizontal strictness

Put E=K_{Z/X}. The maximality argument in Section 5.4 gives

Its concrete input is a domination inequality b'g^*L-f^*H' pseudoeffective for a sufficiently ample base polarization H'. Two generic-fiber independent adjoint sections would, after an ample base twist, push down to an admissible pencil with ratio outside \mathbb C(Y), contradicting the defining maximality. The Jacobian section supplies the nonzero rank-one generator.

Let \alpha\ge0 be the smooth curvature of the original pulled-back metric on g^*L, normalized to represent g^*c_1(L), and let r be its maximum rank. If r=n=\dim Z, then L is nef with positive top self-intersection and hence big, giving the required nonvanishing. Suppose r<n. Choose Kähler forms \omega_Z,\omega_Y representing ample classes. Domination gives b'[\alpha]-[f^*\omega_Y] pseudoeffective for some b'>0. Intersecting with [\alpha]^r[\omega_Z]^{n-r-1}, and using \alpha^{r+1}=0, yields

The integrand is nonnegative, so it vanishes pointwise. At a rank-r point, this says that f^*\omega_Y vanishes on \ker\alpha. A nonnegative Hermitian form has zero mixed pairings against any vector on which its quadratic value vanishes. Hence

At a smooth point this forces r\ge b; rank zero is therefore excluded. The maximum-rank open set meets both the complement of E and the smooth base-change locus. On a smaller relatively compact constant-rank patch there, (6.11) gives \alpha\ge c f^*\omega_Y for some c>0.

Proposition 6.3 now applies with its birational map g:Z\to X and fibration f:Z\to Y. It supplies an integral effective A supported on \mathcal S and a semipositive metric on -K_Y+A, smooth and strictly positive on a nonempty open set, whose weighted density belongs to L^{1+\eta}_{\rm loc}. Lemma 6.5 gives an effective C_A supported on A with W+C_A rationally effective. The return lemma applies to P\sim_{\mathbb Q}g^*L: its horizontal part is nonnegative, its minimum divisor is W, and every component of f^*C_A is exceptional over X. It gives the desired section of a positive multiple of L.

This proves Theorem 5.1 using the original smooth coefficient metric. The determinant and tensor consequences remain Lemma 5.5 and Corollary 5.6. The next section retains a different bigness proof and a fiber-intersection test for horizontal positivity.

7. Jet separation and a nef-and-big decomposition

There is another way to obtain the boundary needed for nonvanishing. A metric that is strictly positive on one open set first separates linearly growing jets at a point. This proves bigness directly by Nadel vanishing. A subsequent multiplier-ideal construction produces a nef and big divisor on a resolution, which can be perturbed by any fixed pseudoeffective class. We prove this base lemma independently, then apply it to an Iitaka fibration. The Iitaka construction also provides a different proof of the horizontal curvature bound.

7.1. A base lemma proved by jets and a nef-and-big decomposition

7.2. The Iitaka base revisited

We apply the jet construction to the Iitaka base already used in Section 5.2. The algebraic return remains Lemma 5.2; the new point is a fiber-intersection proof that the original smooth metric is strictly positive in base directions. We also record the stronger generic-fiber rank statement behind this version of the argument.

Retain X and L=-K_X with its smooth semipositive metric from Theorem 5.1. Write the given effective integral divisors as N_m\sim mL-D for an unbounded set of positive integers m, with D pseudoeffective Cartier, and fix two exponents j<q. Use the Iitaka fibration of N_q constructed in Section 5.2. That construction settles the cases of zero-dimensional and generically finite base. In the remaining case it gives

where A_0 is ample Cartier, u is a positive integer, and I\ge0. The varieties are smooth projective, f has connected fibers, and every source prime mapping into base codimension at least two is \pi-exceptional. Every ratio from a multiple of N_q belongs to \mathbb C(Y), which is relatively algebraically closed in \mathbb C(X). These are precisely the model and field properties established there; no flatness of the final smooth resolution is needed.

Put E=K_{S/X} and D_S=\pi^*N_q+E. Adding positive multiples of E does not change the sections of multiples of \pi^*N_q, because E is exceptional and X is normal. Thus \kappa(D_S)=b and uD_S-f^*A_0 has an effective representative. In fact

For if this rank exceeded one, an ample base twist would yield two sections of \ell D_S+af^*A_0 independent over \mathbb C(Y). Multiplication by the section of a(uD_S-f^*A_0) places them in (\ell+au)D_S. Their ratio is transcendental over \mathbb C(Y). Other sections of a sufficiently divisible multiple recover b algebraically independent base functions through A_0. Products put all these ratios into one multiple of D_S, contradicting \kappa(D_S)=b. The canonical sections show that the rank cannot be zero. At the generic point \eta of Y, this gives h^0(S_\eta,E|_{S_\eta})=1, and \pi^*N_q|_{S_\eta} has Iitaka dimension zero.

The interpolation and vertical-minimum construction of Section 5.2, with these indices, gives

Here R has nonnegative horizontal part and W is pseudoeffective. The rank statement above also gives the interpolation directly: for each given m>q, restrict the two effective divisors (m-j)N_q\sim(m-q)N_j+(q-j)N_m to the generic fiber. Its one-dimensional space of sections makes their horizontal divisors equal. The horizontal slope is therefore nonnegative as m tends to infinity. The principal difference has vertical divisor, so its restriction to the projective normal generic fiber is constant; hence it is pulled back from Y. Taking vertical minima in this equality recovers the pseudoeffectivity of W by the fixed-error limit used above.

Call a base prime missed when every source prime above its generic point is \pi-exceptional. The model property also makes all other components of its pullback exceptional. By Lemma 5.2, it is enough to construct a weighted metric satisfying Theorem 7.1 on -K_Y+A, with A supported on missed primes. We now verify the only new ingredient, horizontal strictness, using intersections on the fibers.

7.3. Horizontal positivity from fiber intersections

Let \widetilde L=\pi^*L and let \alpha be its smooth semipositive curvature form. Since D is pseudoeffective and u\pi^*N_q-f^*A_0 has an effective representative, k\widetilde L-f^*A_0 is pseudoeffective for some k>0. Let J be ample on S, put d_0=n-b, and choose the largest t\in\{0,\ldots,d_0\} such that

on a smooth general fiber. The choice exists because t=0 works. These intersections are constant on the smooth base locus. Semipositivity then makes the restriction of \alpha have rank at most t at every point of every such fiber.

The pseudoeffective comparison and nefness give

Indeed the difference of the b-th powers factors as the pseudoeffective class k\widetilde L-f^*A_0 times a sum of products of nef classes. The final strict inequality is the projection formula and the definition of t. The smooth representative shows that \alpha has rank at least b+t on a nonempty open set.

A vertical null vector for a nonnegative Hermitian form is null for the full form. The vertical null space has dimension at least d_0-t, so the total rank is at most b+t on the smooth locus. On the preceding open set equality holds, and the full kernel is vertical. A smaller constant-rank patch, disjoint from E, therefore satisfies

for some c>0 and a positive base form \omega_Y.

The adjoint rank is one by (7.8). Proposition 6.3 now gives an integral effective A supported on missed primes and a semipositive metric on -K_Y+A, smooth and strictly positive on an ordinary open set, with weighted density in L^{1+\gamma}_{\rm loc} for some \gamma>0. Theorem 7.1 makes W+C_A rationally effective with C_A\ge0 supported on A. Lemma 5.2 applied to R gives a section of a positive multiple of L. This completes the jet-based proof of Theorem 5.1.

8. A Fano-type model of the tensor fibration

The preceding effectivity arguments correct a divisor on a smooth base. Here we instead contract the base until no correction is needed there. The additional geometric conclusion is a Fano-type model of this auxiliary base. Its contracted divisors pull back to exceptional divisors over the original variety, so a principal correction on the model still produces a section on the original variety.

We give a Fano-type proof of the tensor conversion in Corollary 5.6. Its input is a smooth connected projective X with smoothly semipositive L=-K_X, and nonzero sections of (\Omega_X^1)^{\otimes p}\otimes mL for one fixed p\geq0 and unbounded positive m. The distinctive output is the supported birational model of Theorem 8.1. We first construct the auxiliary base and verify the hypotheses of the toroidal volume theorem. We then identify exactly which base divisors its log terminal model can contract. The final passage to an anticanonical section uses the principal divisors themselves, not only their numerical classes.

8.1. A fibration tangent to the null directions

The cases \dim X=0 and p=0 are immediate. Suppose otherwise. Write \mathcal E=(\Omega_X^1)^{\otimes p} and apply the determinant extraction in Lemma 5.5. Retain its determinant line M, rather than its negative: there is a nonzero map

for some s>0, and effective integral divisors

for an unbounded set of positive integers m. The generic wedges used in that lemma are regular because they take values in the saturated subsheaf and its reflexive determinant. This is the determinant method of [23, Lemma 4.1] and [22, Lemma 5.1], with the unbounded exponents supplied here as part of the hypothesis.

Write n=\dim X, let \alpha be the normalized curvature of L, and let \nu be its maximum rank. If \nu=n, then L is nef and L^n=\int_X\alpha^n>0, so L is big and conversion follows. Fix an ample class A and assume \nu<n. We will use

Here is a differential-geometric proof of this equality. For each \epsilon>0, Yau’s prescribed-Ricci theorem [34] gives a Kähler metric in [\alpha]+\epsilon A whose Ricci form equals the prescribed semipositive form \alpha, up to the fixed positive normalization. Kähler symmetry identifies the mean curvature of the tangent bundle with its nonnegative Ricci endomorphism. Thus the mean curvature of every cotangent tensor power, and of its exterior powers, is nonpositive. Consequently

Indeed, a positive degree would allow a metric on M with constant positive mean curvature, by the scalar Poisson equation. The induced mean curvature on \bigwedge^s\mathcal E\otimes M^{-1} would then be negative definite, and the integrated Bochner formula would forbid its nonzero section. Since \alpha^{\nu+1}=0, the coefficient of the lowest possible power \epsilon^{n-1-\nu} is a positive constant times c_1(M)[\alpha]^\nu A^{n-1-\nu} and is nonpositive. The same intersection for N_m equals it by (8.1), and is nonnegative by effectivity. This proves (8.2).

Choose a finite sum D of the N_m, allowing repetitions, whose Iitaka dimension d is maximal among such sums. These dimensions are integers in \{0,\ldots,n\}, so a maximum exists. Choose q>0 for which |qD| has image dimension d. On the open set where \alpha has rank \nu, the differential of this rational map annihilates \ker\alpha. To see this, each member of |qD| has zero intersection with [\alpha]^\nu A^{n-1-\nu}. A smooth local component whose tangent hyperplane failed to contain \ker\alpha would retain rank \nu for the restricted form, and hence contribute positive mass to that intersection. Away from the base locus, a projective hyperplane detecting a nonzero differential on \ker\alpha would produce exactly such a smooth local component. Hyperplanes detect the differential of the map, so the assertion follows. In particular d\leq\nu<n.

Resolve the base ideal by p_1:X_1\to X, with X_1 smooth and the resolution an isomorphism off that ideal, and take the Stein factorization f_1:X_1\to B. The normal projective base B has relatively algebraically closed function field in \mathbb C(X). There are an ample divisor A_B and an effective fixed divisor I with

Every p_1-exceptional prime lies in \operatorname{Supp}I, since its center lies in the base ideal and that ideal has positive order on the prime.

Fix m_1<m_2 among the determinant exponents and put

On any compatible higher model \mu:W\to X, f:W\to Y, with Y birational over B, there are \mathbb Q-principal divisors T_m on Y such that

For the principal difference of

is a rational function on B. Otherwise it is transcendental over \mathbb C(B), by relative algebraic closedness, and adjoining it to the map for |qD| increases the image dimension. The joint map is given by products of sections in the system for the permitted finite sum qD+(m-m_1)N_{m_2}, contradicting maximality. Dividing its principal divisor by m_2-m_1 proves (8.4). If d=0, these functions are constant: N_m=mG+J, so every coefficient of G is nonnegative by letting m\to\infty. This already proves conversion. We henceforth have 0<d<n.

We next establish the adjoint rank required for the volume metric. For each integer \ell>0, the general fiber of f_1 satisfies

If the dimension were larger, generic base change and a sufficiently ample base twist would give sections of \ell I+t f_1^*A_B whose ratio varies on the general fiber. For t\gg\ell, multiply them by the section of (t-\ell)I. They become sections of t p_1^*(qD). The same complete system recovers the base using the sections from |tA_B| multiplied by s_I^t, so its image dimension exceeds d, a contradiction. The Jacobian divisor K_{X_1/X} is effective and bounded by some \ell I. Its nonzero canonical section and (8.5) therefore give rank one for its full direct image. Further smooth modifications do not change this rank: their extra relative canonical divisors are effective and exceptional, and their section pushforwards equal \mathcal O by normality. Thus on any smooth resolution \pi:Z\to X, g:Z\to Y dominating these models,

The Jacobian section identifies g_*(K_{Z/Y}+\pi^*L) with -K_Y on the smooth base-change locus. This is an ordinary rank assertion, involving the entire adjoint space.

8.2. The integrable correction and the model it permits

Apply birational weak toroidalization [1, Theorem 1.1], marking the nonisomorphism locus over X, and equidimensional toroidal subdivision [2, Proposition 4.4]. We obtain \mu:W\to X, f:W\to Y with W normal projective, Y smooth projective, and f surjective, equidimensional and toroidal for strict toroidal embeddings. All modifications are birational; no reduced-fiber alteration is required. The source can be singular, and its resolution is used only for integration. The effective exceptional Weil divisor K_{W/X} is supported in the toroidal boundary.

Choose a global Kähler form \omega_Y on Y. By generic smoothness and cohomology-and-base-change, there is a nonempty Zariski open U\subset Y on which the resolved map g is smooth and its adjoint direct image is the line identified above. The kernel calculation persists off the exceptional loci, since the map from the base to the original image is generically finite. On a small patch above U, in the maximum-rank region and away from the Jacobian divisor, it gives

The second inequality follows from the first by semipositive linear algebra and a lower bound for the nonzero eigenvalues on a smaller patch. Together with (8.6), these verify all hypotheses of the toroidal volume theorem in [29, Theorem 2.1].

Explicitly, for a base prime P and primes S\subset W dominating it, put a_S=\operatorname{ord}_S K_{W/X} and e_S=\operatorname{ord}_S f^*P. The theorem supplies

and a semipositive metric on \Lambda=-K_Y+C whose local squared frame norms are integrable and whose curvature is smooth and strictly positive on a nonempty ordinary open set. Moreover C is effective rational with finite support and f^*C is entirely exceptional over X. The companion theorem proves the all-valuation integrability and the extensions across divisors and codimension two; these conclusions use the precise ordinary rank and patch verified here.

In particular \kappa(Y,C)=0. Indeed, a rational function with poles bounded by a positive divisible multiple of C pulls back to a function with only \mu-exceptional poles. It extends across X by normality and is constant because X is connected projective. Also there is an effective big rational divisor

Apply Corollary 4.2 to the unweighted integrable metric on \Lambda. Its proof first subtracts a small ample rational class while keeping a klt representative, then adds a general representative of that ample class. Thus the boundary here is effective, big and klt, exactly as required by the birational theorem below.

8.3. Returning the correction to the original variety

Apply Theorem 8.1 to (8.8). We now finish this proof of Corollary 5.6. For each base prime P set

These are finite rational sums. The horizontal coefficients of \mu^*G are nonnegative by (8.4) and m\to\infty. At a component attaining the minimum g_P, that same equation and effectivity of \mu^*N_m give

Thus mB_g+B_j+T_m\geq0. Push this inequality by strict transform to Y_0. Principal terms remain principal because \chi extracts no divisors. Dividing numerical classes by m proves that (B_g)_0 is pseudoeffective. Theorem 8.1 supplies a \mathbb Q-principal adjustment making (B_g)_0 effective. Use its same rational functions and coefficients on Y, and call the resulting principal divisor T_*. Then B_g+T_* is effective outside the primes contracted by \chi.

The divisor \mu^*G+f^*T_* has nonnegative coefficient at every prime of W that is not exceptional over X. Horizontals were already treated. Above an uncontracted prime this follows from the defining minimum g_P. Above a contracted prime, every component is exceptional over X, since that prime lies in \operatorname{Supp}C and f^*C is exceptional. Equidimensionality leaves no other vertical primes. Pushing forward to X therefore gives an effective rational divisor. Its class is \mathbb Q-linearly equivalent to G\sim_{\mathbb Q}L, because the base principal adjustment pulls back and pushes forward as a principal divisor. Clearing denominators completes the proof.

The Fano-type conclusion belongs to Y_0, the model of an auxiliary linear-system base. The argument on X uses exactly the supported exceptional correction; it does not assert that X itself is of Fano type.

9. Invariant sections from a curvature-null base

The ordinary conversion theorem does not control characters. We now construct the base from invariant ratios and retain an invariant rational section throughout the return step. The cohomological input comes from the natural anticanonical index. Here the natural linearization on -K_X=\det T_X is the action induced by the differential of the action on X. The subsequent determinant, base, and metric constructions are local to the present argument.

A connected torus acts trivially on ordinary cohomology by isotopy and hence on H^q(X,\mathcal O_X) by Hodge theory. Thus \chi(X,\mathcal O_X) is exactly the zero value in Proposition 9.1. The point case is immediate. For n=\dim X>0, average the metric over the maximal compact torus and write its curvature as \theta\ge0, normalized to represent c_1(L). Let \nu be its maximum rank and let \omega_0 be a Kähler form. The invariant forms supplied by Proposition 9.1 give the answer immediately if their degree is zero. Otherwise determinant extraction as in Lemma 5.5 gives a linearized line A, a nonzero map A\to\bigwedge^e\Omega_X^p, and invariant sections

We will use an intersection consequence of the determinant map, rather than replace this construction by the ordinary conversion theorem.

9.1. The invariant ratio field

Put B_j=A+m_jL. Form the subfield generated by ratios of invariant sections of the same line bundle \sum_j k_jB_j, where k_j\ge0 are integers of finite support. Let F be its relative algebraic closure in \mathbb C(X). Both fields are finitely generated: choose a transcendence basis for the smaller field; the algebraic closure of the associated purely transcendental field in the finitely generated extension \mathbb C(X) is finite, and the intervening algebraic fields are finite as well. The field F is fixed by T. Indeed an element algebraic over the invariant subfield has finite orbit among the roots of a polynomial, and connectedness makes this orbit a point.

Finitely many ratios can be put over a common denominator by multiplication of sections. Thus one bundle B=\sum k_jB_j and invariant sections of B define an image Y_0 with the smaller function field. Normalize in F, resolve the base, and obtain the invariant rational map X\dashrightarrow Y, with \mathbb C(Y)=F and geometrically integral generic fiber. Let H_0 be the pullback of the image polarization; it is big and base point free. On a smooth equivariant graph resolution

there is a nonzero invariant section of \pi^*B-g^*H_0: divide the pulled-back defining sections by the homogeneous coordinates on the image. Hence for a chosen ample H on Y there is an integer b>0 and an invariant section of b\pi^*B-g^*H.

If F=\mathbb C, consider

Its numerator and denominator are invariant sections of the same multiple of B_2, so r_j is constant. Divisor comparison and m_j\to\infty show that \operatorname{div}(\sigma_2/\sigma_1)\ge0. This is the desired invariant section of (m_2-m_1)L. We may therefore suppose \dim Y>0.

There is a horizontal strictness patch on Z. If \nu=n, use any maximum-rank point away from the exceptional and critical loci. If \nu<n, Lemma 9.3 gives c_1(bB)\cdot[\theta]^\nu[\omega_0]^{n-1-\nu}\le0. Since b\pi^*B-g^*H is effective, and all the forms involved are semipositive, it follows that

At a maximum-rank point where \pi is an isomorphism, nonnegative linear algebra implies \ker\pi^*\theta\subset\ker dg. Choose such a point over the smooth base-change locus. Shrinking a constant-rank neighborhood gives

on a nonempty ordinary patch, disjoint from the Jacobian divisor. Rank zero is impossible here when \dim Y>0.

On this equivariant model, with the trivial action on Y, the invariant part of g_*\mathcal O_Z(K_{Z/X}) has rank one. The natural Jacobian section gives rank at least one. If it were larger, global generation after a sufficiently ample base twist would give two invariant sections independent on the generic fiber. Multiplying them by a power of the invariant section of b\pi^*B-g^*H puts them in K_{Z/X}+k b\pi^*B. Birational pushforward, using \pi_*\mathcal O_Z(K_{Z/X})=\mathcal O_X, gives a ratio of invariant sections of kbB not in F, a contradiction. This is invariant rank one; no assertion about the rank of the full direct image has been made.

We have obtained the analytic data: smooth invariant coefficients, invariant adjoint rank one, the natural nonvanishing Jacobian generator, and the horizontal patch (9.3). To obtain unweighted integrability with exceptional support, we now use a separate normal toroidal model. Equivariance of that model is not required because the meromorphic data on X are already invariant.

9.2. A klt model and its unweighted volume

By weak toroidalization and equidimensional subdivision [1, 2], prepare

with W normal and simplicial toroidal, Y smooth, f equidimensional and toroidal, and the p-exceptional locus in the boundary. The base may be modified birationally, and the smooth equivariant graph resolution above is then replaced accordingly. For clarity, the simplicial condition means that the primitive rays of each local toric cone are linearly independent. Such toric charts are finite abelian quotients of smooth toric charts. The simplicial refinement can be made projectively using generic heights on the existing rays. Since the base cones are simplicial and source rays map to base rays or zero, each subdivided cone still maps onto a face of a base cone. Equidimensionality is preserved. Simplicial toroidal charts have quotient singularities; in particular W is \mathbb Q-factorial and klt.

Put D_j=\operatorname{div}(p^*\sigma_j) and V=(D_2-D_1)/(m_2-m_1)\sim_{\mathbb Q}p^*L. Equation (9.2) gives

Thus V has nonnegative horizontal part. Every vertical prime of W dominates a base prime. Its minimum divisor

is pseudoeffective by the coefficientwise finite-error argument following (5.2). On W one has the actual inequality V\ge f^*M.

The smooth metric of L defines a positive volume \mu on X. On a smooth base-change open, let \rho be the coordinate density of g_*\pi^*\mu. This measure can equally be computed on W_{\rm reg}; birational changes affect only sets of measure zero. Its density is the squared norm of the Jacobian adjoint generator divided by the base canonical frame. Compact averaging on the smooth direct image is a holomorphic orthogonal projection onto the invariant line. Its norm is therefore the quotient norm. For positive-dimensional fibers, Lemma 6.1 gives

with strictness on a nonempty ordinary open by (9.3). If the relative dimension is zero, connected general fibers make g birational. Over its isomorphism open the integral metric is simply the transported coefficient metric; its curvature gives the same assertion directly, including strictness on the patch. This use of a smooth coefficient and an orthogonal quotient does not assert positivity of an arbitrary subbundle of a singular direct image.

Write E_W=K_{W/X}\ge0. For a base prime Q, set

This effective rational divisor has finite support, and f^*F_0\le E_W. In fact, if b_Q>0, then e_Db_Q\le1+a_D-e_D\le a_D for every D\mapsto Q. If a prime above Q is nonexceptional over X, its a_D is zero and b_Q=0. Hence f^*F_0 is exceptional over X.

On -K_Y+F_0 consider the weight

initially off the indicated divisors, with q_Q a local equation of Q. At a general point of a prime D attaining the minimum, choose coordinates with f=(z_1^{e_D},z_2,\ldots,z_{\dim Y}). The pulled-back volume density is comparable to |z_1|^{2a_D} times a smooth positive density. Integration of this chart gives

Consequently \Phi is locally bounded above at a general point of every omitted divisor. Plurisubharmonic extension, first across these divisors and then across codimension two, gives a semipositive metric on the rational line -K_Y+F_0, still smooth and strictly positive on an ordinary open.

Its squared frame norm is unweighted locally integrable. To check all valuations, take a log resolution h:W'\to W of the relevant data. Integrating e^{-\Phi} on the base is the same as integrating the pulled-back volume divided by the local divisor factors of f^*F_0. The exponent divisor upstairs is

The first summand has coefficients greater than -1 because W is klt, and the second is effective because E_W-f^*F_0 is effective and rationally Cartier. On a normal-crossings resolution all coefficients are therefore greater than -1, which is exactly the local Jacobian integrability test. This includes exceptional divisors above higher-codimension centers.

Apply Corollary 5.4 to the pseudoeffective M. It gives F_0+\epsilon M\sim_{\mathbb Q}G\ge0 for some rational \epsilon>0. Pullback and the inequality V\ge f^*M make V effective after a principal correction from Y and a divisor exceptional over X. Equivalently apply Lemma 5.2 on a common resolution. The underlying rational section is a power of \sigma_2/\sigma_1 times a base function, so it is invariant. Its extension on X proves Theorem 9.2.

10. A maximal invariant base and signed adjoint metrics

There is another invariant construction when H^q(X,\mathcal O_X)=0 for every q>0. The maximal base is chosen from all invariant rational maps whose polarizations are pseudoeffectively dominated by L. Its distinguishing output is a sequence of signed boundaries, one for every positive adjoint twist. The limit of those boundaries proves the pseudoeffectivity needed for nonvanishing; the metric for a single twist would not suffice.

More precisely, if X is smooth connected projective over \mathbb C with H^q(X,\mathcal O_X)=0 for all q>0, a compact real torus T_c acts holomorphically, and the naturally linearized L=-K_X is smoothly semipositive, this construction gives H^0(X,mL)^{T_c}\ne0 for some m>0. Since H^1(X,\mathcal O_X)=0, the torus preserves the isomorphism class of a very ample polarization and acts in the projective linear stabilizer of its image. A compact torus there lies in an algebraic torus after conjugation; taking its Zariski closure therefore permits equivariant projective graph resolutions. Averaging metric weights gives a smooth invariant coefficient. The zero-index value is one. Proposition 9.1 and Lemma 5.5 supply a linearized A with -A pseudoeffective and sections as in (9.1), unless a scalar invariant section has already been obtained.

10.1. Maximality and all positive adjoint ranks

Consider invariant dominant rational maps to projective bases with trivial action, subject on a resolution \pi:Z\to X, g:Z\to Y to

for some ample H and rational \delta>0. Include the point map. This condition is unchanged by replacing the resolution. It persists under birational modification of the base or its finite Stein refinement: the pullback of an ample divisor under a generically finite map is big and dominates a small ample class. It also persists for the joint image of two such maps, by addition and a product polarization. Choose a map of maximal base dimension, take the relative algebraic closure of its field in \mathbb C(X), and resolve the base. The resulting field is still invariant, as in the finite-orbit argument above. Every other admissible invariant ratio belongs to it: its adjoining cannot increase dimension, and the field is already relatively algebraically closed.

Fix two exponents a<b in the determinant sequence, set m=b-a, and put t=\sigma_b/\sigma_a, an invariant rational section of mL. For every larger exponent i, the ratio

belongs to \mathbb C(Y). It is a pencil in a bundle jL+dA with j,d>0, and -A pseudoeffective makes that pencil satisfy (10.1). Thus the horizontal orders of the \sigma_i vary affinely in i. Their nonnegativity at unbounded i makes D_t=\operatorname{div}(\pi^*t) effective horizontally. If Y is a point, this gives the desired section directly.

On a smooth equivariant graph resolution set E=K_{Z/X}. For every integer k\ge1 the invariant generic adjoint rank is

The rational section s_Et^k is regular on the generic fiber, hence supplies a nonzero invariant element. If the rank exceeded one, global generation after twisting by jg^*H would give two invariant sections in E+km\pi^*L+jg^*H whose ratio is not a base function. Their effective divisors push to a pencil in D_0=kmL+j\pi_*g^*H on X; exceptionality removes E. Pushing (10.1) forward gives an integer b>0 for which bL-D_0 is pseudoeffective. Resolve the pushed-down pencil by \sigma:U\to X and h:U\to\mathbb P^1. Its fixed part F is effective and \sigma^*D_0\sim h^*\mathcal O_{\mathbb P^1}(1)+F. Consequently

This is exactly the domination premise for the new invariant pencil. Its ratio is not a base function, contradicting maximality.

We use a normal \mathbb Q-factorial simplicial toroidal equidimensional model p:W\to X, f:W\to Y, as above, for the divisorial calculations. It need not carry the action. All positivity on invariant summands will instead be proved on the smooth equivariant model, with base change and the ranks (10.2).

10.2. The signed boundaries and their integrals

For a prime D\mapsto Q on W, put

Here a_D\ge1, and a_D=1 for nonexceptional primes. Define

The first minimum records the vertical slope of t; the second includes the Jacobian order relevant to integration. We choose the coefficient \beta_Q(k) of the base correction to satisfy

These inequalities have different roles. Near general points of D and Q, the local map y_1=z_1^{e_D} turns a volume density of order 2(kd_D+a_D-1) into a contribution to the base density with exponent 2((kd_D+a_D)/e_D-1). A prime attaining the minimum therefore gives a lower bound with exponent 2(\lambda_Q(k)-1). Dividing the base density by |y_1|^{2\beta_Q(k)} adds \beta_Q(k)\log|y_1|^2 to its negative logarithm. The first lower bound makes this corrected weight locally bounded above at Q. The divided volume upstairs has exponent 2(kd_D+a_D-1-e_D\beta_Q(k)); the strict upper bound makes it integrable at each prime over Q. The toroidal calculation below will check the remaining valuations. The final upper bound ensures that the section returned to X has no pole at a nonexceptional prime. The comparison with k\gamma_Q will be used after taking the limit of the correction divisors.

All four requirements are met by

The supports lie in one finite set. Since every a_D/e_D>0, both terms in the maximum are strictly below \lambda_Q(k). For a nonexceptional D, both k\gamma_Q and \lambda_Q(k)-1\le kd_D/e_D+1/e_D-1 are at most kd_D/e_D. Finally, \lambda_Q(k)-k\gamma_Q is bounded independently of k, because the minimum is over a fixed finite set. Thus

The chosen lower bound by k\gamma_Q also gives B_1\ge B_\infty. After the metrics show that -K_Y+B_k is pseudoeffective for every k, the displayed limit will imply pseudoeffectivity of B_\infty and hence of B_1.

The pair

is sub-klt. Horizontal coefficients are nonpositive. A vertical coefficient equals e_D\beta_Q(k)-kd_D-(a_D-1)<1. If e_D=1, both terms in the maximum defining \beta_Q(k) give this coefficient at most zero. Thus the positive part is supported on the toroidal boundary. On a simplicial toroidal chart the discrepancy function has positive values 1-b_D on the boundary rays, and every new ray has a positive linear combination of those values. Hence an effective boundary with coefficients below one is klt there. Subtracting an effective rational divisor only improves discrepancies. This proves (10.4), including all divisorial valuations.

Let d\mu be the smooth anticanonical volume on X. On the smooth base-change locus the invariant rank-one generator of K_{Z/Y}+(km+1)\pi^*L is the relative form associated to s_Et^k. The squared norm in its coordinate frame is

Smooth direct-image positivity applies to the smooth invariant coefficient (km+1)\pi^*L. Compact averaging is a holomorphic orthogonal projection, so the invariant line has the quotient metric. The squared frame norm on G_k=-K_Y+B_k is

Near a prime attaining the minimum in \lambda_Q(k), the local map has form y_1=z_1^{e_D} and the numerator volume is comparable to |z_1|^{2(kd_D+a_D-1)} times smooth volume. Therefore

Since \beta_Q(k)\ge\lambda_Q(k)-1, the weight -\log H_k is locally bounded above at general points of all missing divisors. It extends plurisubharmonically there and across codimension two. Thus every G_k is pseudoeffective.

Moreover H_k is unweighted locally integrable. On a log resolution h:W'\to W, the divisor of poles in the integrand is

Its coefficients are below one by (10.4). The normal-crossings Jacobian test and properness prove integrability on the base. This signed boundary calculation is separate from the effective exceptional correction in Section 9.2.

10.3. A global strict metric and the final section

We give the strict-metric construction with its actual sheaf and ideal conditions. From (10.1) choose a singular metric on \pi^*L with curvature at least \delta g^*\omega_H. Average its global weight difference u from the smooth invariant metric over T_c, in the sense of distributions. This preserves the quasiplurisubharmonic lower bound. On (m+1)\pi^*L, add \epsilon u to the smooth coefficient weight. For small \epsilon>0 the resulting metric h_\epsilon satisfies

Small-exponent integrability on compact Z proves the ideal assertion. Consequently the singular adjoint direct-image theorem [31, Theorem 3.3.5] acts on the full ordinary direct-image sheaf. Let U\subset Y be a smooth base-change open on which this sheaf is locally free. Fubini makes the fiber integral norm finite almost everywhere on U.

On U, averaging over the compact torus defines a holomorphic projection P onto the invariant summand. Invariance of h_\epsilon makes P orthogonal almost everywhere, where the integral norm is finite. The dual of the quotient embeds holomorphically in the dual full bundle, and its dual norm is the restricted norm. Logarithms of these dual norms are plurisubharmonic by direct-image positivity. They therefore define the semipositive quotient metric; its rank is one by (10.2). Subtracting a base potential before applying this argument retains the lower curvature bound \epsilon\delta\omega_H. This proves positivity for this specified quotient, without a claim about arbitrary singular invariant subbundles.

For k=1 the integral norm is obtained by inserting e^{-\epsilon u} in the density \rho_1. Since u is locally bounded above, the lower density estimates remain valid up to constants. The same extension across divisors, and then codimension two, gives a metric on G_1=-K_Y+B_1 with a global strict curvature lower bound. Together with the already constructed integrable semipositive metric, strong openness and Hölder’s inequality give a globally strict metric on G_1 whose squared frame norms remain locally integrable. Indeed take a sufficiently small positive fraction of the strict weight; the integrable metric has some L^{1+\eta} room by strong openness, and the new singular weight has small-exponent integrability.

It remains to verify the algebraic premise for the adjoint class. Every G_k/k is pseudoeffective, and G_k/k=-K_Y/k+B_k/k\to B_\infty. The closedness of the pseudoeffective cone gives B_\infty pseudoeffective. Since B_1\ge B_\infty, B_1 is pseudoeffective as well. Choose a small ample rational A' whose smooth curvature can be subtracted from the global strict lower bound on G_1. The resulting metric on G_1-A' remains globally strict and unweighted integrable. Apply Lemma 2.1 with J=0 and a sufficiently large divisible r>1. On its log resolution, the fixed part divided by r, minus the relative canonical divisor, has coefficients below one. A general member of the free part, divided by r, meets this support transversely and has coefficient 1/r<1. It therefore gives an effective klt \Delta_0\sim_{\mathbb Q}G_1-A'. Add a general effective rational representative of A' with small coefficients, transverse on a common log resolution. The resulting \Delta\sim_{\mathbb Q}G_1 is effective, big, and klt. Only the global strict bound and unweighted integrability are used here; the mixed metric need not be smooth on an open set. Now K_Y+\Delta\sim_{\mathbb Q}B_1 is pseudoeffective, so big-boundary nonvanishing [8, Theorem D] gives

for some N>0 and h\in\mathbb C(Y)^*, after applying Lemma 3.1. The rational section t^Nf^*h has no pole at horizontal primes. At a vertical prime nonexceptional over X this follows from \beta_Q(1)\le d_D/e_D in (10.3). Equidimensionality handles every such prime on W. It therefore extends to a nonzero section of NmL on X. Both t and the base function are invariant, proving invariant nonvanishing under the stated cohomology-vanishing hypothesis by this second construction.

11. Compact isometric monodromy and descent

Let X be a smooth connected projective complex variety whose anticanonical bundle has a smooth Hermitian metric of semipositive Chern curvature. The finite-cover structure theorem supplies a compact factor to which the preceding invariant results apply. There are two legitimate orders of operation: descend invariant scalar sections, or descend twisted forms before applying an ordinary conversion theorem. We give the transport in both cases, since the second does not assume an equivariant conclusion from Theorem 5.1.

The finite-cover structure theorem [27, Theorem 2.1] supplies a connected finite étale cover \widehat X\to X whose universal cover has the holomorphic isometric splitting

Here R is a product of compact simply connected Ricci-flat factors, F is a product of the remaining compact projective irreducible factors, and the deck group projects to a full translation lattice on \mathbb C^a. Its action on the compact factors lies in a compact connected torus of holomorphic isometries, and the flat and Ricci-flat factors have invariant nowhere-vanishing canonical frames. Moreover H^q(F,\mathcal O_F)=0 for q>0, and L_F=-K_F is smoothly semipositive. Empty products and point factors are allowed. These geometric and canonical-frame properties are supplied by the cited theorem and its proof. The separate anticanonical nonvanishing theorem of [27, Theorem 1.1] is not used here.

For scalar descent, apply Theorem 9.2 to F, or the compact-torus construction of Section 10. The compact action is contained in an algebraic torus: H^1(F,\mathcal O_F)=0 preserves a very ample polarization, and a compact abelian projective linear group is conjugate into a diagonal torus. Thus a positive power of L_F has a section fixed by the projected deck action. Pull it to \widetilde X and multiply its coefficient by the matching inverse powers of the invariant canonical frames on \mathbb C^a and R. The resulting anticanonical section is deck-invariant and nonzero, so it descends to \widehat X. If F is a point, begin with its constant section. The finite étale section norm then gives a nonzero anticanonical multiple on X.

For the second order of operation, Proposition 9.1 gives a fixed p and invariant nonzero sections

at unbounded positive m, since the actual zero index equals one. The pullback map for differential forms on the product is injective. Tensoring the coefficient with the (m+1)-st inverse powers of the canonical frames on the other factors gives a nonzero form with coefficient K_{\widetilde X}^{-(m+1)}. It is deck-invariant: translations preserve the flat frame, compact monodromy preserves the Ricci-flat frame, and u_m is invariant on F. It therefore descends to a nonzero section of

These holomorphic sections are algebraic because \widehat X is projective. If p=0, we have a scalar section already. Otherwise Corollary 5.6 converts the descended forms on the actual projective cover to an ordinary anticanonical section. The invariant information was used before this ordinary conversion, not inferred from it afterward.

For clarity, the norm does not need a Galois cover. If \nu:\widehat X\to X has degree d and s\in H^0(\widehat X,-kK_{\widehat X}) is nonzero, étaleness gives -kK_{\widehat X}=\nu^*(-kK_X). Locally the product of s over the d sheets is invariant under their permutation, so it glues to a section of -dkK_X. At a general point every factor is nonzero, so the norm is nonzero. This is [27, Lemma 2.2], here with all its hypotheses visible.

11.1. Recovering an invariant factor section

There is also a useful converse transport after ordinary conversion. It identifies precisely the analytic role of isometric monodromy.

12. Extremal characters and invariant tensors

The tensor conversion proved above is ordinary: its input tensors need not be invariant. This permits another way to handle compact monodromy. We first construct invariant tensors on the compact factor, then descend them before applying Corollary 5.6. The character construction below uses the extreme terms of the full equivariant Euler characteristic. It is independent of the weight-zero index calculation used in the preceding applications.

We use the left action on sections: a group element transports a fiber value and changes its argument by the inverse action. Thus a tangent weight a induces conormal weight -a. This convention fixes both the orbit-degree identity and the expansions below. The use of torus characters follows the equivariant viewpoint in [24]; the present argument produces tensors to be converted on the total space, without a semiampleness assumption on the compact factor.

12.1. Descent before ordinary conversion

We finish this method by specifying the tensor descent. Use the finite-cover structure statement as in Section 11: on the universal cover of a connected finite étale cover \widehat X\to X there is an isometric holomorphic product

Here R is the compact Ricci-flat product, F is projective with no positive-degree holomorphic forms, and -K_F is smoothly semipositive. The chosen deck subgroup acts by translations on the flat factor, fixes the canonical frame on R, and acts on F through a connected compact torus of holomorphic isometries. Point factors are allowed. These are the geometric structure and frame assertions, independent of the anticanonical nonvanishing conclusion.

For completeness, the compact action has the algebraic realization needed by Proposition 12.1. Since H^1(F,\mathcal O_F)=0, it preserves a very ample line up to isomorphism and hence acts projectively in its complete linear-system embedding. The representation is continuous, as one sees by tracking a projective frame of points on the embedded variety. A connected compact abelian subgroup of the projective linear group is conjugate into a diagonal torus. Its Zariski closure is an algebraic torus preserving F; quotient by its ineffective kernel if necessary. A diagonal projective torus has a linear lift after fixing one diagonal coordinate, so an ample linearization is available. The action on -K_F remains the intrinsic tangent-determinant action.

Pull each invariant tensor u_k from the proposition to \widetilde X and use the cotangent inclusion from the product projection to F. If \omega_{\mathrm{flat}} and \omega_R are the invariant canonical frames of the other factors, then

is nonzero and deck-invariant. It descends to \widehat X and is algebraic by projectivity. Corollary 5.6 now gives a nonzero anticanonical plurisection on this actual projective finite cover. Finally the finite étale norm recalled in Section 11 gives a nonzero positive anticanonical multiple on X. Thus the only invariant objects needed for this method are the tensors before descent; the conversion and its Fano-type auxiliary model are ordinary.