Anticanonical nonvanishing from smooth semipositivity

Proof. We first obtain the product and its cohomological properties. We then choose a finite-index deck subgroup and identify its compact action with an algebraic torus action.

The metric splitting and the compact factors. The curvature form of the given metric on K_X^{-1} is a smooth semipositive representative of its first Chern class, with the usual normalizing factor. Yau’s prescribed-Ricci theorem therefore gives a Kähler metric on X whose Ricci form is that curvature form [19]; this application is also stated explicitly in [8, p. 218]. In particular its Ricci tensor is nonnegative. The structure theorem of [8] gives a holomorphic isometric decomposition

where each M_i is compact and simply connected and the nonflat factors are irreducible in the real de Rham decomposition. The compactness of these factors is the Cheeger–Gromoll input in the structure theorem; the holomorphic decomposition comes from the parallel complex structure. Group the Ricci-flat factors into F and the remaining factors into Z.

Here is the Bochner argument for the part of this decomposition used below. On a compact Kähler manifold with nonnegative Ricci tensor, a holomorphic p-form u is parallel. Moreover, if \lambda_1,\ldots,\lambda_n\geq0 are the Ricci eigenvalues in a unitary frame and u=\sum_{|J|=p}u_J\,dz_J, the integrated Bochner formula forces the pointwise vanishing of

These are the Bochner consequences used in [8, §2]; equivalently see the formula in [5, (1.6)]. In particular, contraction of u with a positive Ricci eigendirection is zero. If u\ne0 on an irreducible factor, its contraction kernel

is a proper parallel complex subbundle. At any point where the Ricci tensor is nonzero, that kernel is also nonzero. Its underlying real subbundle contradicts irreducibility. Thus every non-Ricci-flat factor has no nonzero holomorphic form of positive degree. The decomposition of exterior forms on a product and Kähler Hodge symmetry give (2.2). Rational connectedness of these factors, including the irreducible compact Hermitian-symmetric case, is the additional conclusion of [5, Theorem 1.4(a) and §4]; it is not inferred merely from the vanishing of forms.

The canonical Chern connection of the Ricci-flat factor F is flat. Since F is simply connected, parallel transport gives a nowhere-zero parallel holomorphic frame \eta_F of K_F. The metric induced on K_Z^{-1} has semipositive curvature, since the Ricci tensor of the product restricts to that of Z.

Projectivity of each compact factor follows directly from that of X. Fix points in the other factors and compose its slice in \widetilde X with the covering map to X. This holomorphic map has injective differential. Pulling back a positive Hermitian metric on an ample line bundle of X therefore gives a positive line bundle on the compact factor. Kodaira’s embedding theorem makes that factor projective [12]. The argument requires the slice map to be immersive, not injective.

Choosing the deck subgroup. Let \Gamma=\pi_1(X) act on \widetilde X by deck transformations. These are holomorphic isometries. Uniqueness of the de Rham decomposition implies that, after a finite-index passage removing permutations of its factors, every deck transformation is a product map. Put M=F\times Z. Projection of this deck subgroup to the Euclidean isometry group is discrete and has finite kernel. Indeed, for a fixed bound on the displacement of 0\in\mathbb C^a, a sufficiently large closed Euclidean ball B satisfies

for every deck transformation within that bound. Since B\times M is compact, proper discontinuity allows only finitely many such transformations. This proves both assertions. The Euclidean image is cocompact, by compactness of the quotient of the full product. Bieberbach’s theorem supplies a finite-index translation lattice in this image, as in [8, §3(ii)].

For completeness, the finite kernel does not obstruct choosing an actual lattice subgroup of the deck group. Above the translation lattice we have an extension

with N finite. Pass to the centralizer of N in \Gamma_1; this has finite index because \operatorname{Aut}(N) is finite. Its kernel N' is central, and its image \Lambda' is a sublattice of finite index in \Lambda_1. Choose lifts g_1,\ldots,g_{2a} of a lattice basis \lambda_1,\ldots,\lambda_{2a} of \Lambda'. Their commutators belong to N'. If e>0 is divisible by the orders of all elements of N', then

The commuting elements g_i^e generate a free abelian subgroup: a relation among them projects to a relation among the independent vectors e\lambda_i. Its image is e\Lambda', so this subgroup has finite index in the original deck group. In case a=0, the deck group is finite and we instead take its trivial subgroup.

We next eliminate the action on F. The holomorphic isometry group of a compact simply connected Ricci-flat manifold is finite. To see this, its Lie algebra consists of Killing fields. The integrated Bochner identity makes each such field parallel. Its dual one-form is closed and, by simple connectedness, exact. An exact parallel one-form on a compact manifold is zero: a primitive attains a maximum, where its differential vanishes. Thus the compact isometry group has zero-dimensional Lie algebra and is finite. Passing to the kernel of the action on F preserves a finite-index lattice subgroup.

The remaining action on Z has commuting image in its compact holomorphic isometry group. Its closure is a compact abelian Lie group. Its identity component is a torus and its component group is finite. Pass once more to the preimage of that identity component. The new image is dense in the identity component, since the latter is open in the old closure. Denote this connected closure by T, the deck lattice by \Lambda, and its action on Z by \rho. This proves (2.3).

The corresponding quotient S is a connected finite unramified holomorphic cover of X. The pulled-back positive line bundle makes S projective, and the covering is a finite étale morphism of projective varieties. We have now accounted for every part of the deck action. In particular \eta_F is invariant because the action on F is trivial, while \eta_0=dz_1\wedge\cdots\wedge dz_a is invariant under translations. The determinant of the product tangent decomposition gives (2.5), with its natural equivariance.

The algebraic torus. It remains to put the compact action on Z in the algebraic category. The point case is immediate, so suppose \dim Z>0 and choose a very ample line bundle A on Z. By (2.2), the exponential sequence makes

injective. Every element of the connected group T is isotopic to the identity and hence acts trivially on integral cohomology. Consequently t^*A\simeq A for every t\in T. The complete linear system of A therefore defines a canonical projective representation

preserving the embedded variety Z. Choices of isomorphisms t^*A\simeq A change the associated linear maps only by scalars, so this projective representation is well-defined. It is continuous: choose a projective frame from the nondegenerate irreducible embedded variety; a projective transformation is determined continuously by the images of that frame. Such a frame exists by first choosing a projective basis and then a point outside its finitely many coordinate hyperplanes.

Write r=\dim H^0(Z,A). To diagonalize the compact projective action, take its compact preimage under the finite covering \operatorname{SL}_r(\mathbb C)\to\operatorname{PGL}_r(\mathbb C). The identity component of this preimage is abelian and surjects onto T: its Lie algebra maps isomorphically to the abelian Lie algebra of T, and its image is an open and closed subgroup of T. Averaging a Hermitian form makes this connected preimage unitary; its commuting matrices can then be diagonalized simultaneously. Thus, after conjugation, T lies in a diagonal algebraic torus in \operatorname{PGL}_r(\mathbb C).

Let G be its Zariski closure there. It is connected, since a connected T cannot meet distinct components of an algebraic group without being disconnected. A connected algebraic subgroup of a diagonal torus is an algebraic torus. The projective stabilizer of Z is Zariski closed, so G preserves Z and acts algebraically. Differentiation gives the natural algebraic linearization on K_Z^{-1}. The induced finite-dimensional representation on H^0(Z,K_Z^{-m}) is algebraic. The stabilizer of any section is Zariski closed, and Zariski density of T proves (2.4). All the constructions respect point factors and the stated canonical-frame conventions.□

Proof. Take an analytic open set U\subset X on which A has a frame e and the cover splits into d sheets. On these sheets write s=f_i\nu^*e for holomorphic functions f_1,\ldots,f_d. Define the local norm by

This expression is unchanged by a permutation of the sheets. If the frame is replaced by e'=u e, the functions become f_i'=f_i/u; their product changes by u^{-d} and the new tensor frame changes by u^d. The expressions therefore glue to a holomorphic section of A^{\otimes d}.

Since S is smooth and connected it is integral. The zero set of the nonzero section s is thus proper and has dimension less than \dim S (or is empty). Its image under the finite map is a proper closed subset of X. Away from that image every sheetwise factor is nonzero, so the norm is nonzero. Projectivity identifies this holomorphic section with an algebraic section. Finally, K_S=\nu^*K_X because \nu is étale; applying the result with A=K_X^{-m} proves the last assertion. In dimension zero the same construction is the product of nonzero constants.□

Proof. First replace S_0 birationally by a projective model of its function field and compose the rational map with that replacement. This preserves the generic fiber and any trivial base action. Resolve the graph of the resulting rational map, starting above the prescribed model if one has been given. Flatten the resulting projective morphism by a modification of its base, using the strict transform [17]. Resolve the modified base and take the main component of the base change, namely the closure of the component over the common dense open. Before taking that component, the flat family has all fibers of the generic dimension d-s; base change preserves this. A closed component has fiber dimension at most d-s. Normalize the main component to obtain \bar Y. Normalization is finite, so this upper bound persists. The dimension inequality for a dominant map gives the reverse bound for every irreducible fiber component. Thus \bar f is equidimensional. Resolve \bar Y to obtain Y; no equidimensionality is asserted for f.

For an invariant rational map, use its invariant graph. The strict transform and main component are invariant, the action lifts uniquely through normalization, and characteristic-zero equivariant resolution supplies the smooth source. All base modifications have trivial torus action, so the resulting maps are equivariant.

If a prime Q has image of codimension at least two in S, its generic fiber over its image has dimension at least (d-1)-(s-2)=d-s+1, contradicting equidimensionality. Hence the only vertical divisors lie above base divisors. At general points of Q and T, both divisors are smooth, Y\to\bar Y is an isomorphism, and the induced map Q\to T is a submersion. A local equation of T pulls back to t^e times a unit. Take a holomorphic eth root of that unit and absorb it into t; choose the remaining base coordinates among coordinates on Q. This gives (3.2). Generic smoothness and constructibility ensure that these neighborhoods occur over a dense open of T, while avoiding any specified proper closed subset of Q.□

Proof. Set n=\dim Z and b=\operatorname{rank}Q. Resolve the rational fibration using Lemma 3.1. Thus there are morphisms

where Y,S are smooth projective, \bar Y is normal, \bar\pi and r are birational, and \bar f is equidimensional with connected general fibres. Their dimension is n-b. The foliation is the saturated relative tangent sheaf at the generic point.

Pull back s_D as a top form with coefficients, and denote the result by

Its divisor is K_{Y/Z}+\pi^*D; the relative canonical divisor is the effective birational Jacobian. Change of variables off the exceptional locus shows that the density of \eta for \pi^*h_J has finite total mass. This assertion concerns almost-everywhere densities, so no pointwise product on the metric’s polar set is needed.

Over a coordinate chart in the base, let ds be a nonvanishing frame of K_S. On a smooth fibre Y_s, the restriction of \eta/f^*ds is a section of K_{Y_s}+\pi^*J|_{Y_s}. It is nonzero on general fibres, and Fubini’s theorem gives finite squared norm on almost every such fibre. Berndtsson–Păun’s relative Bergman kernel theorem therefore supplies a semipositive metric on

[3, Theorem 0.1]. Over the smooth-fibre locus, in a holomorphic frame its weight \psi is the logarithm of the supremum of the squared absolute coefficients of fibre sections having squared norm at most one. Only sections integrable for the restricted twisting metric enter that supremum.

We now compare this metric with the line bundle in the statement. Remove from Z a closed subset of codimension at least two so that \pi is an isomorphism over the remaining open set and the sheaves \mathcal F,Q are locally free there. At the generic point of every divisor of this open set the conormal inclusion is

For a horizontal divisor it has full rank without a divisorial zero: the restriction of f to that divisor is generically submersive in characteristic zero. Every other divisor under consideration maps onto a base divisor. Indeed its strict transform on \bar Y is a prime divisor, and equidimensionality excludes a base image of codimension at least two.

Fix such a vertical divisor E. At its general points there are coordinates on Y and S in which

To obtain these coordinates, take a point where the reduced map E\to\{s_1=0\} is submersive, write f^*s_1=z_1^e u with u a unit, and absorb a local e-th root of u into z_1. Complete the other pulled-back base coordinates to a coordinate system. In this chart the saturated conormal bundle is spanned by dz_1,\ldots,dz_b. The determinant of its displayed inclusion is e z_1^{e-1}. Consequently its vanishing order is exactly e-1.

This same integer occurs in the Bergman metric. Choose a frame v of \pi^*J and use

as the frame of K_{Y/S}+\pi^*J. If a\xi is a fibre section, then, up to a constant of absolute value one from the order of wedge factors, its coefficient as an actual fibre top form is

This distinction between relative and fibre coefficients is also the one used in the definition of the relative Bergman kernel [3, Section 1].

Choose two concentric coordinate polydiscs whose closures lie in this chart. For each fixed (z_1,\ldots,z_b) with z_1\ne0, the larger polydisc in the free variables z_{b+1},\ldots,z_n lies in the same smooth nearby fibre. Its radius can be fixed independently of the base point. The twisting frame norm has a uniform positive lower bound there. The holomorphic submean inequality on that polydisc, for points in the smaller one, therefore gives

where c,c'>0 do not depend on the fibre section or the nearby base point. For squared norm at most one, this implies

The estimate holds on the smooth-fibre locus; any exceptional almost-everywhere qualifications disappear for the psh representative by the submean inequality.

On the submersion locus where \pi is an isomorphism we have K_{Y/S}+\pi^*J=D-f^*K_S. The determinant computation above says that a frame of D-P maps to z_1^{e-1} times a nonvanishing multiple of a frame of D-f^*K_S. The induced metric on D-P thus has weight

up to a pluriharmonic frame term. This weight is psh off E and is locally bounded above by (4.3); hence it extends psh across E. At a horizontal divisor there is no ramification correction, and the original Bergman metric already extends. The coordinate argument applies over a dense open subset of each divisor. The remaining excluded subset has codimension at least two in the smooth variety Z, across which psh weights extend uniquely. The extensions respect changes of frame and therefore define a semipositive metric on the global line bundle D-P.□

Proof. Assume first that \dim Z>0, fix \alpha, and set \delta=D\cdot\alpha\geq0. Semipositivity of J gives

Let \lambda_1>\cdots>\lambda_t be the slopes of the Harder–Narasimhan quotients of T_Z, with positive ranks r_1,\ldots,r_t. Suppose, for a contradiction, that \lambda_t<-\delta. The total degree of the pieces with nonpositive slopes then satisfies

By (4.6), some piece has positive slope. Consequently the positive part \mathcal F of the filtration is nonzero and proper. It is saturated, and

The bracket modulo \mathcal F is an \mathcal O_Z-linear morphism \bigwedge^2\mathcal F\to T_Z/\mathcal F. Since the target is torsion-free, it factors through A=(\bigwedge^2\mathcal F)/\mathrm{torsion}. When \operatorname{rank}\mathcal F>1, the tensor slope formula, duality, and the quotient inequality give \mu_{\min,\alpha}(A)\geq2\mu_{\min,\alpha}(\mathcal F)>0. The maximum slope of the target is nonpositive, so this morphism vanishes. If \operatorname{rank}\mathcal F=1, then A=0. Thus \mathcal F is a foliation. Campana–Păun’s algebraicity criterion now shows that it is algebraically integrable [6, Theorem 1.1].

Lemma 4.1 gives c_1(T_Z/\mathcal F)\cdot\alpha\geq-\delta. Its left side is exactly the sum in (4.7), a contradiction. This proves (4.4). In particular the argument covers the cases in which the positive part would be zero or the whole tangent sheaf: the assumed failure of the inequality has already excluded both cases.

By duality and tensor-product compatibility,

for k>0. A nonzero map from M is injective, because its target is torsion-free. Thus (kD-M)\cdot\alpha\geq0 for every movable \alpha, and divisor–curve duality proves (4.5). For the sheaf E, the same inequality applied to its slope gives c_1(E)\cdot\alpha\leq krD\cdot\alpha, which proves the determinant assertion. Determinants are taken reflexively; removing codimension-two singular loci does not change their first Chern classes.

If k=0, the target is \mathcal O_Z. A nonzero map from M provides a nonzero section of -M, and a positive-rank subsheaf of \mathcal O_Z has rank one with effective negative determinant. These arguments also cover dimension zero. In that dimension there is no nonzero cotangent tensor for k>0, and no minimum slope of a zero tangent sheaf is invoked.□

Proof. The class of L+D is on the boundary of the closed pseudoeffective cone, whose interior is the big cone. A supporting hyperplane and cone duality give a nonzero real movable class \alpha annihilating it. Both L and D pair nonnegatively with \alpha, so they pair with it individually to zero. Theorem 4.2 gives the tangent inequality and M\cdot\alpha\leq0. Effectivity of each N_i gives the reverse inequality through N_i\cdot\alpha=M\cdot\alpha. Finally an interior point of a full-dimensional closed convex cone pairs strictly positively with every nonzero member of its dual cone; hence no indicated combination is big.□

Proof. We first normalize \rho so that a divided section is regular and nonzero on at least one component over each base prime. We then use its adjoint integrals twice: the zeroth integral gives the volume metric, and the high moments give the bounded metric. The support of the first correction will make the final lifting possible.

If S is a point, every prime of \bar Y is horizontal. Thus \rho is regular by normality and descends as a nonzero section of kL on the smooth variety X. Take A trivial and D_S=0; the metric assertions on a point and the lifting assertion follow. We assume \dim S>0 below.

Divisorial normalization. For a prime divisor T\subset S, define

where Q ranges over the prime divisors of \bar Y mapping onto T. These are finitely many components for each T, and the denominators are positive integers. Only finitely many c_T are nonzero. Replace \rho by a power and k by the corresponding multiple to make every c_T integral, and set A=\OO_S(\sum_Tc_TT). The rank condition (5.1) persists: the new adjoint twists form a subsequence of the old ones and still include j=0.

Divide \rho by the pullback of the tautological rational section of A, obtaining a rational section e of k\bar\pi^*L-\bar f^*A. Its order is nonnegative at horizontal primes by hypothesis and at vertical primes by (5.2). Equidimensionality ensures that every vertical prime maps onto a base prime. Thus these are all prime divisors of \bar Y, and normality makes e regular. For each base prime T, a component attaining the minimum in (5.2) has order zero in e. We also write e for its pullback to Y.

The fiber integrals and their curvature. Pullback of the J-valued top form includes the Jacobian:

Change of variables gives finite total mass for \mu_Y. In particular this assertion concerns measures and does not require a pointwise value for a singular metric on its polar set.

On a coordinate chart U\subset S, take a canonical frame ds of K_S and a frame v of A. The section e_v=e f^*v belongs to k\pi^*L over f^{-1}(U). For j=0,1,2,\ldots, set

The section u_j belongs to K_{Y/S}+\pi^*(J+jkL). On a smooth fiber it is a top form with coefficients in the indicated twisting line; (5.3) pairs that form with its conjugate to give a positive density. Properness, regularity of e_v, and boundedness of h_L’s weights bound |e_v|^2 on inverse images of relatively compact base charts. Since V_0 is the density of the finite measure f_*\mu_Y, all V_j are finite and positive almost everywhere on a suitable smooth-fibration open.

For clarity, changes of frames have the following effect:

This will give weights on two different lines: -K_S at j=0, and A after division by j and passage to the limit.

The relative Bergman theorem [3, Theorem 0.1] applies to the semipositive twisting metric \pi^*(h_Jh_L^{jk}): (5.3) supplies a nonzero square-integrable adjoint section on almost every smooth fiber. The theorem produces a semipositive relative Bergman metric. On a dense base open where base change holds and (5.1) has dimension one, u_j generates the adjoint section space. For almost every such fiber V_j<\infty, so its one-dimensional holomorphic adjoint space is also its full space of square-integrable sections: every section is a scalar multiple of u_j. Consequently the Bergman weight, in a holomorphic frame upstairs, is

almost everywhere, where \widehat u_j is the coefficient of u_j. Off the zero locus of that coefficient, its logarithm is pluriharmonic, so subtracting it leaves a psh function upstairs. In a local product chart, Fubini’s theorem gives almost all constant slices on which the displayed identity holds almost everywhere. Restriction to any of these slices gives a psh representative of \Phi_j on the base. For fixed base frames, these representatives agree almost everywhere on overlaps, hence everywhere, and therefore patch.

The base-change open can depend on j, so we now put all these weights on one fixed open S^\circ. Choose S^\circ such that f is smooth there and every fiber has a point where both \eta_Y and e are nonzero. A psh weight is locally bounded above; therefore its squared frame norm has a positive local lower bound, even when the metric is singular. On a small product chart around such a point, this lower bound for h_J, together with the bounds for h_L, gives

locally on S^\circ, with positive C_1,C_2 independent of j. Removable singularities extend each psh representative across its additional exceptional base loci inside S^\circ. The union of the exceptional measure-zero sets for the countably many j still has measure zero; hence all moment comparisons can be used simultaneously. Almost-everywhere upper bounds also hold for the psh representatives by the submean inequality.

We have now obtained all the moment weights on a fixed open set. To continue the induction, both kinds of weight must extend to the boundary. Their estimates use different components of a singular fiber.

The zeroth moment: finite volume with supported correction. Call a base prime T bad if every component of \bar f^*T is \bar\pi-exceptional or lies in \Supp(\bar\pi^*D). Only finitely many primes are bad: each is the image of a member of the finite list of exceptional and boundary divisors on \bar Y.

Take a prime T in the complement of S^\circ, and a component Q above it on \bar Y. At general points of its strict transform in Y, there are coordinates

Indeed, the map on the divisors is generically smooth, and a root of a nonvanishing unit absorbs the unit in the first coordinate. Write the coefficient of \eta_Y as t^q times a unit at such a point, with q\ge0. Division by ds contributes t^{-(\ell-1)}, so the relative top-form coefficient has magnitude comparable to |t|^{q-\ell+1}. The metric lower bound and integration over a fixed patch in the free fiber coordinates give

almost everywhere near a general point of T.

If T is not bad, choose Q nonexceptional over X and outside \Supp(\bar\pi^*D). Then the birational Jacobian and s_D are generically nonzero on Q, so q=0. Thus -\log V_0 is locally bounded above there. At the bad primes choose sufficiently large nonnegative integral coefficients for D_S. Equation (5.6) makes the weights

locally bounded above at general points of every boundary prime. Here the hat denotes the coefficient in a local frame of \OO_S(D_S). These weights are psh off the boundary, extend across its general divisorial points, and then extend across the remaining analytic set of codimension at least two. The frame rule (5.4) gives a metric on -K_S+D_S.

Pairing (5.7) with s_{D_S} cancels the added factor. The resulting adjoint measure equals f_*\mu_Y almost everywhere, so has finite total mass. We have obtained the volume metric and the required support of its correction.

High moments: bounded weights on A. For l=1,2,\ldots, put on S^\circ

where the star denotes upper-semicontinuous regularization in fixed local frames. Equation (5.5) gives uniform local upper bounds. The regularized suprema are therefore psh and decrease.

Their limit does not collapse to -\infty. On any relatively compact base chart U, including charts meeting the boundary, properness and bounded weights of h_L give a single constant C_U>0 with

The first bound holds on the whole inverse image of U: take a finite cover of its compact closure by line-bundle frames. For almost every s\in U\cap S^\circ, V_0(s) is finite and positive. Dividing (5.9) by j after taking negative logarithms gives \Psi\ge-\log C_U. Thus \Psi is psh. Its frame transformation is exact, although that of each normalized moment has a vanishing error. On a relatively compact overlap, (5.4) changes \Phi_j/j by -\log|g|^2+(\log|h|^2)/j. The last term has absolute value at most C/l for every j\ge l. The two tail suprema, and then their upper-semicontinuous regularizations, therefore differ from one another plus -\log|g|^2 by at most C/l. Taking the decreasing limit gives precisely \Psi\mapsto\Psi-\log|g|^2, the weight rule for A.

To extend \Psi, now choose a component Q attaining the minimum in (5.2). This need not be the component used for the zeroth moment. Its defining property is that e is nonzero generically on Q. The local norm of e_v is therefore bounded below on a suitable patch. The same power-map calculation as in (5.6), with the factor e_v^j, gives

for a nonnegative integer b_T independent of j. For example, increasing a nonnegative integer above (q-\ell+1)/\ell suffices after shrinking the coordinate chart. The absence of zeros of e on the chosen component is what prevents an additional exponent proportional to j.

Shrink the chart so that |s_1|<1, and decrease C_1 if needed so that C_1\le1. Put

For j\ge l, the negative logarithm of (5.10) gives

The right side is continuous off T, so it also bounds the upper-semicontinuous regularization of the tail supremum there. Letting l\to\infty in (5.8) gives \Psi\le-\log C_2 throughout the punctured chart. Hence \Psi is locally bounded above near general points of each boundary prime, and extends psh across them and then across codimension two. The lower bound in (5.9) was uniform on the whole base chart, so the extension retains it. The extended psh weight is locally bounded above as well. Thus the metric on A has locally bounded weights on both sides.

Lifting the section. We have the two metrics needed on the base. It remains to verify that the divisor used for finite volume creates only permitted poles on X. Regard a nonzero section of aA+bD_S as a rational section of aA with poles only on D_S. Pull it to the normal equidimensional model \bar Y and multiply by e^a. The resulting nonzero rational section of ak\bar\pi^*L has poles only above bad primes. By their definition every such prime divisor of \bar Y is exceptional over X or lies above D.

At every prime of X outside D, the strict transform in \bar Y therefore has no pole. The rational section downstairs extends there; exceptional divisors give no additional obstruction on the smooth, hence normal, variety X. Its remaining finitely many pole orders are cleared by multiplication by s_D^c for a sufficiently large integer c\ge0. This gives a nonzero section of akL+cD, as required.□

Proof. Hodge symmetry gives H^q(Z,\OO_Z)=0 for q>0, and hence \chi(\OO_Z)=1. Riemann–Roch makes \chi(K_Z+mL) a polynomial in the integer m, whose constant term is

by Serre duality. It is therefore nonzero for all but finitely many positive integers. Some fixed q has H^q(Z,K_Z+mL)\ne0 for unbounded positive m.

For each such m, the metric h_L^m has trivial multiplier ideal: its weight is locally bounded, so every holomorphic germ is locally square integrable. Hard Lefschetz with multiplier ideals [9, Theorem 2.1.1] gives a surjection

by wedging with the qth power of a Kähler form. Choose nonzero maps -mL\to U, where U=\Omega_Z^{d-q}, for an increasing sequence of these exponents.

At the generic point of Z, take the subspace of U_{\C(Z)} spanned by all chosen images after a given cutoff. These subspaces form a decreasing chain in a finite-dimensional vector space. Their dimensions eventually stabilize, so the subspaces themselves stabilize to a nonzero subspace W. Let \mathcal G\subset U be its saturated extension. It is coherent: saturate the images of finitely many maps spanning W. Every map in a sufficiently far tail factors through \mathcal G, because U/\mathcal G is torsion-free and the composite vanishes generically.

Put r=\operatorname{rank}\mathcal G and M=(\bigwedge^r\mathcal G)^{**}. This is a line bundle on smooth Z. For every cutoff, choose r generically independent maps from the tail. Their exterior product is a nonzero section of M+mL, where m is the sum of their exponents. As the cutoff tends to infinity, so does m. The maps and their determinant extend across codimension two into the line M. Selecting an increasing subsequence gives the sections in the statement.

For d-q>0, antisymmetrization embeds U in (\Omega_Z^1)^{\otimes(d-q)}. The tensor slope formula and the assumed tangent inequality give \mu_{\max,\alpha}(U)\le0, and therefore M\cdot\alpha\le0. If d-q=0, then U=\OO_Z and its nonzero saturated subsheaf is \OO_Z itself; thus M=\OO_Z and the same conclusion holds.□

Proof. Consider the nonempty complete linear systems of

Their degrees against \alpha are nonpositive. A divisor whose complete system defines a generically finite map is big. But a big class is an interior point of the pseudoeffective cone, and so pairs strictly positively with every nonzero member of the dual cone. Thus every system in (6.2) has image dimension less than d.

Choose B_0 in this collection with maximal image dimension s, and let V be the closed image of its rational map. This maximum exists because the possible dimensions are integers between 0 and d-1. Let K be the relative algebraic closure of \C(V) in \C(Z); it is a finite extension of \C(V), since \C(Z) is a finitely generated field extension. Take a smooth projective model S of K, with a morphism to V after resolution. Its dimension is s<d, and the rational map Z\dashrightarrow S has connected general fiber.

Every ratio of two nonzero sections in any one system (6.2) belongs to K. Indeed, combine that system with |B_0| by taking products of sections. The resulting linear subsystem of |B+B_0| gives the joint rational map. If its section ratio were transcendental over \C(V), the joint image would have dimension at least s+1, contrary to the choice of B_0. The ratio is therefore algebraic over \C(V) and lies in its relative algebraic closure. We will use this section-ratio property both for interpolation and for rank one.

Resolve |B_0| before applying Lemma 3.1. The resulting smooth model carries an effective divisor R_0 and a line bundle H_0 on S such that

Here H_0 is the pullback of \OO_V(1). If s>0 it is big, since S\to V is generically finite. All additional modifications preserve the displayed effective decomposition by pullback.

Set k=m_2-m_1>0 and take \rho=\bar\pi^*(s_2/s_1). For i>2, the numerator and denominator in

are sections of the same line bundle of the form (6.2): both have M coefficient m_i-m_1 and L coefficient m_2(m_i-m_1). Thus (6.4) is a nonzero element of K. It has order zero at every horizontal prime Q of \bar Y, so

The left side is bounded below independently of i, since each \bar\pi^*s_i is regular. As m_i\to\infty, the right side could have this lower bound only if \ord_Q(\rho)\ge0. This proves (i). If S is a point, every prime is horizontal; normality makes \rho regular, and birational pushdown gives the claimed section on Z. In this case the rank assertion also follows from the section-ratio property. Indeed, the adjoint space in (6.1) identifies with H^0(Z,D+jkL): the Jacobian divisor is effective and exceptional, so a rational section allowed to have poles only there extends downstairs across codimension two on smooth Z. The section s_D\rho^j is nonzero, and any two sections of D+jkL have ratio in K=\C by (6.2). Hence this space has dimension one for every j\ge0, including j=0.

Assume s>0. We now verify (6.1) in every nonnegative degree. The identity

shows which exceptional divisor is involved. Over the generic point of S, a base canonical frame is immaterial. The Jacobian section, s_D, and \rho^j give a nonzero section of the right side. To justify regularity, (i) implies that \rho is regular on \bar Y over a dense base open: remove the images of its finitely many vertical polar divisors. Its pullback to Y is then regular there. Thus the adjoint space has dimension at least one, also for j=0.

Take any two nonzero sections of that generic-fiber space. Choose a rational frame of K_S and regard them, by (6.6), as rational sections of K_{Y/Z}+\pi^*(D+jkL) with no horizontal poles. Their finitely many vertical polar divisors can be cleared by one effective Cartier divisor C on S: enclose their images, including any of codimension at least two, in a Cartier divisor and increase its multiplicities so that f^*C dominates both polar divisors. Since H_0 is big, there are an integer a>0 and a nonzero section of aH_0-C. Multiplication by this section and by s_C makes both rational sections regular after the common base twist af^*H_0.

Now multiply both by the canonical section of aR_0. Formula (6.3) produces two regular sections of

with the same ratio as the original pair. They descend to sections of D+jkL+aB_0 on Z. In detail, K_{Y/Z} is effective and exceptional, and \pi_*\OO_Y(K_{Y/Z})=\OO_Z: a rational function with poles only on exceptional divisors is regular away from a codimension-two subset of smooth Z and hence regular everywhere. Apply this fact after trivializing the line bundle on Z. The descended line belongs to (6.2); the section-ratio property therefore places their ratio in K=\C(S). Any two nonzero vectors in the generic-fiber section space are thus proportional over its ground field. This proves dimension one for all j\ge0, without discarding the zeroth degree.□

Proof of Theorem 1.2. We induct on d=\dim Z. In dimension zero Z is a point, every line bundle is trivial, and the conclusion holds with a=1,b=0. Assume d>0 and suppose that no aL+bD, with integers a>0 and b\ge0, has a section.

The bundle L is pseudoeffective because it has a semipositive metric. The class of L+D is therefore pseudoeffective. It is not big: otherwise some positive multiple of L+D would have a nonzero section. Corollary 4.3 gives a nonzero movable class \alpha with L\cdot\alpha=D\cdot\alpha=0 and \mu_{\min,\alpha}(T_Z)\ge0. Apply Lemma 6.1 to obtain the fixed line M and unbounded sections. Then apply Lemma 6.2.

If its base is a point, the resulting section contradicts our assumption. Otherwise Proposition 5.1, applied to that prepared diagram and the two original metrics, supplies a line A and an effective integral divisor D_S on the smaller smooth projective base S. The line A has a semipositive metric with locally bounded weights, while -K_S+D_S has a semipositive metric with finite distinguished adjoint volume.

The no-forms condition also passes to S. A holomorphic form on S pulls back along the dominant rational map on its domain of definition. Properness of S extends the map at the generic point of every divisor of Z, so that domain can be taken to have codimension-two complement. The pulled-back form extends across that complement because \Omega_Z^q is locally free. Dominance in characteristic zero makes pullback injective. Hence H^0(S,\Omega_S^q)=0 for every q>0.

Since \dim S<d, induction produces integers a>0,b\ge0 and a nonzero section of aA+bD_S. The lifting conclusion of Proposition 5.1 gives a nonzero section of akL+cD on Z for some c\ge0. Its exponent of L is positive, so this contradicts the assumption and completes the induction.□

Proof. The normalized line A is given the trivial action. In (5.2) only base divisors are used, so dividing the invariant \rho by their rational section gives an invariant e. The sections u_j in (5.3) are invariant as well.

On a dense smooth-fibration open where the adjoint direct image is locally free and its formation commutes with base change, Berndtsson’s theorem [2, Theorem 1.2] gives its smooth L^2 metric semipositive curvature. Its hypotheses hold because the total space Y is projective, hence Kähler, and each twisting metric \pi^*(h_Jh_L^{jk}) is smooth and semipositive.

Average the holomorphic group action on this vector bundle over the compact torus. This gives a holomorphic projection onto its invariant subbundle, which has rank one by assumption. The fiber integral is invariant under the group, so the projection is orthogonal. The metric on the invariant line is therefore its quotient metric and is semipositive. Since u_j is its nonzero local generator, -\log V_j has the required psh representative.

This is the only positivity step that changes. The fixed-open estimate (5.5), the zeroth-moment extension (5.7), the high-moment envelope (5.8), and both bounds through codimension two now apply exactly as in the ordinary proof. They give the two asserted base metrics. Every base section is invariant because the base action is trivial. The rational lifting multiplies its pullback by e^a, so remains invariant. With D=0, the support condition permits only exceptional poles, and the final extension on X is an invariant holomorphic section.□

Proof. If Z is a point the natural action on its canonical line is trivial, so the constant section proves the claim. Otherwise replace T by its image in \operatorname{Aut}(Z). The no-forms hypothesis and Hodge symmetry give H^1(Z,\OO_Z)=0. The algebraic-torus argument in the proof of Theorem 2.1 applies to this action: connectedness preserves the first Chern class of a very ample line, the vanishing of H^1(\OO_Z) makes that line invariant up to isomorphism, and its projective representation embeds T in a compact diagonal torus. Its Zariski closure G is an algebraic torus acting on Z, and natural sections are T-invariant exactly when they are G-invariant.

Average the given metric over T by averaging local weights, using the natural linearization. Differences of metric weights are global functions, so these averages patch to a smooth invariant metric; its curvature remains semipositive. Take J=L and D=0. The canonical section 1 of K_Z+L=\OO_Z is invariant for the natural action, and its metric pairing is a smooth positive volume form of finite total mass on compact Z.

If the general G-orbit has dimension zero, connectedness of G makes its action trivial: the orbit map is constant on a dense open and hence on Z. Theorem 1.2, with D=0 and J=L, then gives the required section. Suppose henceforth that the general orbit has dimension r>0.

Take the Rosenlicht rational quotient [18, 1]. Its function field is \C(Z)^G, and a dense invariant open has geometric fibers equal to orbits. Because G is connected, its general orbit is geometrically integral; thus the invariant field is relatively algebraically closed in \C(Z) and the quotient has connected general fiber. Choose its prepared equivariant model

with trivial base action. The notation r_Y for the resolution avoids confusion with the orbit dimension r. The generic fiber has dimension r and a geometrically dense G-orbit. After shrinking a base open, f is smooth there, its fibers have a dense orbit, and \pi agrees birationally with the original quotient construction.

We construct the relative section required for invariant transfer. Choose r elements of the Lie algebra of G whose fundamental vector fields v_1,\ldots,v_r are generically independent. Such a choice is possible because the general orbit has dimension r. The vector fields are invariant, since G is commutative, and tangent to the fibers of f. On the smooth-fibration open their wedge is a regular, generically nonzero section of -K_{Y/S}. Tensor it with the pullback of a rational frame of -K_S, regular after shrinking the base open. This gives an invariant anticanonical section on Y over that open. The determinant of d\pi is the natural morphism

Its image is a nonzero invariant rational section of \pi^*L, regular over the same open.

This rational section also defines a section \rho of \bar\pi^*L on \bar Y with no horizontal poles. Indeed, r_Y is an isomorphism at the generic point of each prime divisor of normal \bar Y. Every horizontal prime meets the base open on which the constructed section is regular. The birational identification of function fields therefore gives nonnegative order there.

The generic-fiber adjoint spaces are finite-dimensional \C(S)-vector spaces carrying algebraic representations of the split torus G_{\C(S)}. They decompose into character spaces. A character of G is trivial on the Zariski-dense subgroup T exactly when it is the zero character. Consequently their T-fixed and G-fixed subspaces agree, not merely those of sections on Z itself.

For each j\ge0, on the generic fiber the section

is a nonzero invariant section of K_{Y/S}+\pi^*((j+1)L). The form \pi^*1 includes the effective birational Jacobian, and \rho has no horizontal pole, so this is regular. Any two nonzero invariant sections of this line have an invariant rational function as their ratio. The equivariant birational maps and the defining property of the quotient give

so this ratio belongs to \C(S). Their space has dimension one over \C(S), for every j\ge0 including j=0.

Proposition 7.1 now applies with k=1, with powers subsequently taken there to clear the vertical minima. It supplies a line A and effective divisor D_S on S, with a bounded semipositive metric on A and a finite-volume semipositive metric on -K_S+D_S. Holomorphic forms on S pull back injectively to Z, as in the last part of Section 6; hence S has no positive-degree holomorphic forms. Theorem 1.2 gives a nonzero section of aA+bD_S with a>0,b\ge0. Invariant lifting produces a nonzero invariant section of a positive power of L on Z. Since D=0, every permitted pole upstairs is exceptional over Z and disappears on its normal pushdown. When S is a point the same conclusion is the point-base case of Proposition 7.1. This completes the proof.□

Proof of Theorem 1.1. Use Theorem 2.1 to choose the finite étale cover \nu:S\to X and the universal product \widetilde X=\C^a\times F\times Z with its lattice deck action. The compact factor Z satisfies the hypotheses of Theorem 7.2; let s_Z\ne0 be an invariant section of K_Z^{-m} for some m>0. If Z is a point, take m=1 and s_Z=1.

Using the product identification (2.5), form

where pullbacks by the product projections are understood. The first two frames are invariant under the deck lattice and the last factor is invariant under the closure of its action. The product is therefore invariant and descends to a nonzero holomorphic section of K_S^{-m}. It is algebraic by projectivity. Finally Lemma 2.2 produces a nonzero section of K_X^{-m\deg\nu}. The exponent is positive, as required.□