Cohomological transfer and equivariant anticanonical sections

Proof. Embed the projective graph over Y in \mathbb P^N\times Y. Over the flat open set the fibers define a map to the projective Hilbert scheme [16, Theorem 3.2 and the Hilbert-scheme specialization after Proposition 3.8]. Resolve the closure of its graph, also resolving the base, and pull back the universal family. Flatness over an integral base embeds each affine coordinate ring in its generic localization. The generic fiber is integral, so these coordinate rings have neither nilpotents nor zero divisors. Thus the pulled-back family is integral and is the closure of the original family over the flat open set. Its map to the graph is birational.

Constancy of the Hilbert polynomial bounds the dimension of every fiber above by the generic relative dimension. At any fiber component the height bound for the ideal generated by local parameters of the smooth base gives the opposite bound. This proves purity of fiber dimension. Normalization is finite, so fiber dimensions and dimensions of images of components are preserved. A prime divisor mapping into codimension at least two in Y' would have dimension at most \dim Y'-2+(\dim X-\dim Y')=\dim X-2, a contradiction.

For connectedness apply Stein factorization to the normal total space. Its finite intermediate base has function field the algebraic closure of \mathbb C(Y') in \mathbb C(X); the asserted field condition and normality of Y' make the finite birational morphism an isomorphism. The same reasoning applies after resolution. In the equivariant case, the family over the original open set is stable under the action. Its closure and normalization are therefore stable, and equivariant resolution in characteristic zero supplies the last assertion.□

Proof. In the invariant case average a smooth semipositive metric on L over the maximal compact torus. The chosen linearization of P differs from the natural one by a character, which is unitary on this compact torus, so the averaged metric is invariant for both actions.

The rank assumption at k=0 forces the generic fiber to be geometrically connected. Indeed its constant functions, multiplied by s_E, inject into that adjoint space. In the invariant case the connected torus acts trivially on the finite Stein base, so all these constants are invariant. Their dimension is therefore one. Smooth general fibers are consequently connected and irreducible.

For every prime divisor B of Y, put

Properness and equidimensionality supply divisorial lifts of B; all vertical prime divisors lie over base primes. Only finitely many c_B are nonzero. Clear denominators and use the resulting integral divisor as Q. Sending its rational canonical section to the powered s gives a regular bundle map by normality. Its effective Cartier zero divisor G omits at least one component above every base prime. In the invariant case all these divisors are invariant, the base line carries trivial linearization, and the map is to b\mu^*P. In an eigensection version, one instead twists the base linearization by the section character.

For a local frame e of Q, denote its image by s_e and put

This is continuous. Properness gives upper semicontinuity. For lower semicontinuity use that an equidimensional holomorphic map from a pure dimensional normal space to a smooth base is open. To verify openness at a point, cut the source by as many general local hyperplanes as the fiber dimension, isolating the point in its fiber. The slice germ has dimension at least that of the base and is finite over that smooth base germ. Its image is therefore the entire base germ. This can be done inside any prescribed source neighborhood.

The locus Z_0 where M_e vanishes is algebraic closed: its complement is the constructible image g(W\setminus\operatorname{Supp}G), and continuity makes Z_0 closed in the classical topology; a constructible classically closed subset is Zariski closed. By the omitted-lift property, Z_0 has codimension at least two.

We show first that -\log M_e is plurisubharmonic on a dense open. Work on a fixed smooth-fibration open for f, where the canonical exceptional section and s_e are not identically zero on any fiber component. Take a local frame \eta of K_Y. The section

generates the rank-one adjoint space at the generic point. In the shifted equivariant case its coefficient is D_V+kbP_V; as a metrized ordinary line it is still (kb+1)D_V. Berndtsson’s smooth direct-image theorem [5, Theorem 1.2] applies on the bundle and base-change locus with smooth semipositive coefficient (kb+1)D_V. In the invariant case the invariant line is an orthogonal quotient of the full direct image: the compact torus acts unitarily, and the sum of its nonzero-weight spaces is its orthogonal complement. It therefore has semipositive curvature. This quotient argument, rather than positivity of arbitrary sublines, is what permits taking invariants.

For k\ge1 the resulting plurisubharmonic function is

where \mu_y comes from s_E/f^*\eta with the metric of D_V. It is smooth, nonnegative, of full support, and independent of k. Each integral is smooth positive on our fixed open, so the curvature inequality extends there from its possibly k-dependent base-change open. Normalized to probability measures, its 1/k-powers increase to M_e. They converge locally uniformly: a point just below the positive fiber maximum has a neighborhood of uniformly positive \mu_y-mass on nearby fibers; compactness supplies finitely many such neighborhoods, while bounded total mass gives the upper estimate. Thus (2) converges locally uniformly to -\log M_e.

By continuity this weight extends plurisubharmonically across the bad analytic loci wherever M_e>0. It cannot tend to +\infty across Z_0 of codimension at least two. Through a point of Z_0 choose a small analytic disc whose boundary misses Z_0. Nearby parallel discs have their boundaries in a fixed compact set off Z_0. Since Z_0 has codimension at least two, a dense set of these discs misses it entirely. The maximum principle therefore bounds the weight uniformly at their centers, and continuity off Z_0 supplies the same bound there. A zero of the continuous M_e would contradict this bound. Hence M_e>0 everywhere, and the weight is continuous plurisubharmonic. Its transformation rule under a change of frame is exactly the metric transformation rule for Q.□

Proof of Theorem 1.1. The common ratio field. Put n=\dim X and write N_i=M+iP. For each i, ratios of invariant sections of the same positive multiple of N_i form a field Q_i. For p<q<b in I, the identity

holds with linearizations. Multiplication by suitable powers of the fixed s_p,s_b puts any ratio for N_p or N_b into Q_q. This retains the ratio itself: for a ratio in degree aN_p, multiply its numerator and denominator by s_p^{a(b-q-1)}s_b^{a(q-p)}, and use the analogous formula for a ratio in aN_b. Taking a further index beyond any prescribed two interior indices proves that all Q_i with i>\min I coincide; denote this field by Q.

Fix the first two indices p<q. For b>q the ratio

belongs to Q. If \operatorname{trdeg}_{\mathbb C}Q=0, it is constant. The section divisors are affine in i, and their slope is effective because the indices are unbounded. Thus s_q/s_p is a regular invariant section of (q-p)P.

Otherwise put e=\operatorname{trdeg}Q>0. Choose e independent ratios to define a map to \mathbb P^e, resolve its graph and take Stein factorization. The Stein field is the relative algebraic closure in \mathbb C(X) of the field of the chosen ratios, a finite extension of that field. It contains Q, since Q is algebraic over the chosen ratio field, and is relatively algebraically closed in \mathbb C(X). The connected torus acts trivially on the Stein base because its orbits lie in finite fibers over a base with trivial action. As before take a smooth projective base Y, a normal equidimensional W\to Y with a birational morphism W\to X, and an equivariant smooth resolution V\to W, with maps \mu:W\to X, f_W:W\to Y, and total maps \pi:V\to X and f:V\to Y. Put P_W=\mu^*P. The map f has connected fibers by the Stein construction and normality of Y. Write L'=\pi^*L and P'=\pi^*P.

For each interior index i and every divisor H' on Y,

Indeed the chosen independent ratios belong to Q_i. Put their numerators and denominator in one common degree by products of invariant sections. On the resolved graph they have a common invariant factor whose line is that multiple of N_i minus the pullback of \mathcal O_{\mathbb P^e}(1). The latter line is big on Y, so a sufficiently large multiple dominates H'.

A regular relative section. We next remove the vertical poles while preserving the character. Let D_i be the section divisor on W. Since (3) comes from Y, its horizontal divisor is zero. The affine relations show that the horizontal part of (D_q-D_p)/(q-p) is effective. Normalize the vertical part by subtracting the minimum coefficient divided by pullback multiplicity above each base prime, and let B be the resulting rational base divisor. There is an integer k>0, divisible by q-p and clearing the coefficients of B, for which

is regular. Normality checks regularity in codimension one, and equidimensionality accounts for every vertical prime. Above every base prime \gamma is nonzero at the general point of at least one component. We also write \gamma for its pullback to V.

All adjoint ranks, including zero. The ratio field must also control the fiber integrals. Let E=K_{V/X}\ge0. Its Jacobian section 1_E is invariant for the natural linearization on L', irrespective of the character shift in P'. For all j\ge0,

The section 1_E\gamma^j gives the lower bound over the generic fiber. If the rank were larger, globalize two invariant sections u_1,u_2 independent over \mathbb C(Y) after a twist f^*(aH), where H is ample. Choose i,b\in I with b>i+jk. Multiply u_1^k and u_1^{k-1}u_2 by s_i^k\gamma^{b-i-jk}. Both results are sections of

with ratio u_2/u_1. Apply (4) at index b to cancel this base twist by the same invariant multiplier in the numerator and denominator. Removing the exceptional twist kE and pushing to X leaves two invariant sections of an equal multiple of N_b. Normality extends across the image of the exceptional locus. Their ratio must be in Q\subset\mathbb C(Y), contrary to independence. This proves (6).

The bounded metric and the horizontally positive metric. We now construct the two metrics needed for interpolation. Average a smooth semipositive metric h_0 on L over the maximal compact torus. It is also invariant for P, because the two linearizations differ by a character, unitary on that compact torus. Equation (4) and pseudoeffectivity of -M give L'-\eta f^*H pseudoeffective for some \eta>0 and ample H. Taking a positive current, averaging, and mixing a small positive fraction of its potential with h_0 gives an invariant singular metric h on L' with

Skoda integrability and compactness give the last equality. The metric is bounded below by a positive constant times h_0 on compact coordinate patches: its potential relative to the smooth metric is locally bounded above.

Apply Proposition 2.2 with the normal model W, the chosen linearization P, and the relative section \gamma of kP_W-f_W^*(kB). Equivalently use the rational section (s_q/s_p)^{k/(q-p)} before normalization. Equation (6) gives every invariant adjoint rank, the Jacobian has its natural invariant action, and the averaged smooth metric is invariant also for P. The vertical minima defining kB are exactly those in that proposition. Its smooth coefficient is L'+jkP', ordinary (1+jk)L', so the orthogonal quotient argument applies without any singular-subbundle claim. We obtain a continuous, in particular locally bounded, semipositive metric on kB.

A fixed ideal on the base. The continuous base metric just obtained controls the varying power. The singular degree-zero metric will supply the positive constant term. For j=0 use the singular coefficient metric h. The singular adjoint direct-image theorem [27, Setup 3.2.1 and Theorem 3.3.4] applies over the smooth, locally free base-change locus. Its generic multiplier-ideal inclusion is an isomorphism by (7); fiber integrability is an almost-everywhere assertion by Fubini. The invariant summand is an orthogonal holomorphic quotient for the compact-invariant metric. Thus its line metric has semipositive curvature. Subtract a local potential for \delta\omega_Y from the coefficient weight to see that its curvature is in fact at least \delta\omega_Y.

In a frame \xi of -K_Y, denote the squared norm of 1_Ef^*\xi by F_\xi(y). With base volume in the dual frame, this is the pushforward density of the integrable absolute adjoint form 1_E with metric h. Hence F_\xi is locally integrable on the whole base chart. We next extend its negative logarithm with an explicit exceptional correction.

Boundary extension with exceptional support. For a prime boundary divisor P choose a prime Q\subset V mapping onto it, of pullback multiplicity l and Jacobian order a=\operatorname{ord}_Q(E). At general points write f=(z_1^l,z_2,\ldots,z_e). Integrating on one coordinate patch, using h\ge c h_0, gives almost everywhere

The exponent includes the transverse Jacobian factor |z_1|^{-2(l-1)}. If a nonexceptional component dominates P, choose it for the estimate. Then a=0 and the coefficient of P in the correction is zero. Otherwise every component dominating P is exceptional; choose a nonnegative integer coefficient at least (a+1)/l-1. Summing over the finitely many boundary primes gives D_*\ge0 with f^*D_* exceptional over X.

For the last assertion one must also check primes mapping to codimension at least two on Y. The equidimensional normal model W has relative dimension n-e, so for Z\subset Y of codimension at least two,

Any prime of V over Z therefore maps to codimension at least two on W and on X. Every component dominating a base prime in \Supp D_* is exceptional by the rule defining that support. These two checks account for every component of f^*D_*.

On -K_Y+D_* define the metric by |\xi1_{D_*}|^2=F_\xi on the dense open. Its ordinary-frame weight is -\log F_\xi+\log|g_{D_*}|^2. Estimate (8) bounds this weight above at general boundary points. Subtract a local smooth potential for \delta\omega_Y. The resulting plurisubharmonic weight extends across each boundary divisor away from a codimension-two analytic subset. Hartogs extension for plurisubharmonic functions then fills that subset. Restoring the smooth potential gives a metric on all of Y with curvature at least \delta\omega_Y. If \mathcal J is its multiplier ideal, then

since multiplying the metric coefficient by |g_{D_*}|^2 recovers the locally integrable density F_\xi.

Interpolation and return to X. Tensor by the bounded semipositive metric on jkB. For every j\ge0 the multiplier ideal remains the same \mathcal J, which is not assumed trivial. Nadel vanishing yields

Put \mathcal F=\mathcal O_Y(D_*)\otimes\mathcal J. Its Euler characteristic after twisting by jkB is a polynomial in j. The inclusion (9) gives \mathcal O_Y\hookrightarrow\mathcal F, so at zero this polynomial equals h^0(Y,\mathcal F)>0. A nonzero section therefore exists for some j>0. Pull it back and multiply by \gamma^j to obtain an invariant section of jkP'+f^*D_*. Removing the exceptional twist and pushing down gives the required section of jkP.□

Proof. The polynomial in Proposition 4.1 is not zero. A fixed cohomological degree therefore has nonzero invariant cohomology along an unbounded sequence. Average a Kähler form and the metric weights over the maximal compact torus, and denote the resulting curvature of L by \theta. The smooth semipositive hard Lefschetz theorem [12, Corollary 2.1.2], applied to mL=K_X+(m+1)L, gives the asserted forms. Concretely, if u is harmonic of type (n,q) with values in (m+1)L, Bochner–Kodaira gives

Both terms are nonnegative. Applying the complex-linear star on form indices to u therefore gives a holomorphic form of type (n-q,0). The construction is equivariant. For q=0 it already yields invariant sections of mL; for q=n the holomorphic form itself is a section of (m+1)L. If r=\max\operatorname{rank}\theta=n and q>0, the curvature term makes this form vanish on a nonempty open set, hence everywhere, so this case cannot occur.

The determinant extraction follows the method of [18, Lemma 5.1] and [19, Lemma 4.1]. We spell out the unbounded-twist step, which here also applies when L is numerically trivial. Choose a rational frame of L and divide each form by the corresponding power of that frame. Their span over \C(X) is a subspace of the generic fiber of \Omega_X^{n-q}, of some rank e. Changing the frame rescales each vector by a nonzero rational function, so this subspace is intrinsic. Saturate its top exterior power in \bigwedge^e\Omega_X^{n-q} and call the resulting rank-one subsheaf M. It is reflexive on smooth X, hence a line bundle. The generic span is torus-stable, so its saturation carries the induced linearization.

Choose finitely many of the original forms as a generic basis. Every wedge of e original forms lies generically in M tensored by the sum of their powers of L; saturation makes it a regular section of that line. In the fixed basis, one coordinate of the unbounded family is nonzero for unbounded m. Wedge those forms with the other e-1 fixed basis forms. These wedges are nonzero and invariant, and their twist exponents are m+1 plus a fixed sum. They give the required sections with a_i\to\infty.□

Proof. Every element of the connected torus is homotopic to the identity, so its action on singular cohomology is trivial. By Hodge decomposition its action on every H^q(X,\mathcal O_X) is trivial. Consequently the invariant index at zero equals \chi(X,\mathcal O_X)\ne0. Lemma 4.2 supplies either an invariant section directly or invariant sections of M+a_iL with a_i\to\infty and a nonzero map from M into a positive tensor power of \Omega_X^1. The cotangent-tensor theorem [18, Theorem 4.1], applied to the smooth zero-boundary pair with nef -K_X, says that -M is pseudoeffective. Apply Theorem 1.1 with P=L naturally linearized.□

Proof. For every Kähler class G, Yau’s prescribed-Ricci theorem [30] gives a Kähler metric in G with Ricci form \theta. Kähler curvature symmetry identifies the mean curvature of T_X with Ricci; the induced mean curvature of \bigwedge^e\Omega_X^p is nonpositive. If \deg_G M>0, solve a scalar Laplace equation to give M a metric of positive constant mean curvature. Its inverse tensor the cotangent construction then has negative mean curvature, contradicting the Bochner formula for its nonzero section. Thus c_1(M)G^{n-1}\le0. Put G=[\theta]+t[\omega], divide by t^{n-1-r}, and let t\downarrow0. Since \theta^{r+1}=0,

The same intersection for M+aL is nonnegative by effectivity and smooth semipositivity, and the contribution from aL is zero. It is consequently zero. Each component has zero intersection separately; the nonnegative form \theta^r\wedge\omega^{n-r-1} therefore vanishes on its smooth locus, which gives the last assertion.□

Proof. Białynicki-Birula’s theorem [7] gives positive normal \lambda-weights at S. The first nonzero conormal jet of an eigensection subtracts a nonnegative sum of these weights from the line’s fiber weight, proving the section bound. For an ample line, generation of a sufficiently high power gives the fixed-fiber weight bound. Apply this to bU+A, for an ample linearized A, and let b\to\infty to obtain the bound for nef U.

To verify the full-character identity, evaluate the equivariant fixed-point formula [3, 4] at a\lambda(t) as t\to\infty, divide by t^{m\langle\mu,\lambda\rangle}, and keep a generic in T. The inverse normal Euler factors are

They tend to one at S. Every other fixed component has a negative normal weight, hence an inverse factor tending to zero, and its line weight is no larger. Truncating in the ordinary cohomology degree gives the limit a^{m\mu}\chi(S,mU|_S). Since the index is a finite Laurent polynomial, existence of this limit excludes higher \lambda-weights. The limit is precisely the sum of its terms on the weight hyperplane with pairing m\langle\mu,\lambda\rangle; varying a identifies that sum with the single character m\mu and the stated coefficient. For the trivial action the identity is immediate.□

Proof. Take a generic integral cocharacter \lambda with X^\lambda=X^T. A linearized projective embedding shows that only finitely many walls must be avoided. Białynicki-Birula’s attracting-cell theorem [7] gives a unique source component S with open attracting stratum as t\to0; all its normal \lambda-weights are positive. Projection of that stratum to S gives a dominant rational map X\dashrightarrow S. Pullback of forms is injective and extends across the codimension-two indeterminacy locus on smooth X. Thus S also has no positive-degree holomorphic forms and \chi(S,\mathcal O_S)=1.

Let \mu be the full T-character of L|_S. Apply Lemma 5.1 with U=L.

The right side of (10) is polynomial in m with value one at zero. For infinitely many m>0, one fixed cohomological degree has a nonzero weight-m\mu class. Equivariant hard Lefschetz with coefficient (m+1)L produces, for a fixed p, nonzero eigenforms

Use an invariant Kähler form and the maximal compact torus to preserve all character spaces.

Use the determinant construction in Lemma 4.2: the generic span has rank r, and its saturated determinant is a linearized line M_0\subset\bigwedge^r\Omega_X^p. Wedge forms with unbounded m against r-1 fixed complementary basis forms. If the indices in one such wedge are m_1,\ldots,m_r, its twist degree and character are

Thus there are sections of M_0+iL of weight (i-r)\mu for unbounded i. If p>0, the cotangent tensor theorem [18, Theorem 4.1] makes -M_0 pseudoeffective; for p=0, M_0=\mathcal O_X as an ordinary line bundle. Let P be L with its natural linearization shifted by -\mu, and let M be M_0 with its induced linearization shifted by r\mu. These changes leave the ordinary line bundles unchanged, so -M is pseudoeffective in both cases above. The eigensections just constructed are invariant sections of M+iP, because

Theorem 1.1 yields a nonzero section

in the natural linearization.

There are finitely many source weights as the generic cocharacter varies. Zero lies in the convex hull of the actual section weights k_\mu\mu. Otherwise strict separation gives a generic integral cocharacter pairing negatively with all of them. At its own source, the weight \mu is a sum of tangent weights of positive pairing, or the empty sum for a trivial action, contradicting strict negativity. The weights are integral, so a convex combination giving zero may be chosen rational. Clear its denominators and multiply the corresponding powers of the nonzero v_\mu. The product is a nonzero invariant section in positive degree. For the trivial torus the same argument starts directly from \chi(X,\mathcal O_X)=1 with \mu=0.□

Proof. The input is the injectivity theorem of Fujino–Matsumura [14, Theorem A]. On a compact Kähler manifold it applies to a possibly singular coefficient metric and a smooth semipositive multiplying line bundle S, provided the former curvature dominates a positive multiple of the latter. Multiplication by any nonzero section of S is injective on canonical cohomology with the coefficient multiplier ideal. Here we use S=g^*(mA); the comparison constant can be \epsilon/m, and the ideal is trivial.

We include the reduction to base cohomology, following the slicing and Leray argument in [14, proof of Proposition 1.9, Step 3]. Induct on \dim Y. Choose m large enough for Serre vanishing for all the finitely many sheaves R^jg_*\mathcal F, where \mathcal F=\mathcal O_V(K_V+N). A sufficiently general divisor C\in|mA| is smooth, has smooth inverse image V_C, contains no associated point of these direct images, and contains the image of no component of V. Its restricted coefficient metric has trivial multiplier ideal. To see the last assertion without assuming analytic singularities, pull the locally integrable metric coefficients to the incidence variety of the basepoint-free system on V. That incidence variety is a projective-space bundle over V. Fubini’s theorem gives integrability on almost every smooth member, simultaneously on a finite coordinate cover. This full-measure condition meets the required algebraic open conditions. Restricting the curvature inequality on these members gives the same hypothesis over C.

Adjunction and multiplication by the cutting section give

The corresponding long direct-image sequence breaks into short exact sequences: multiplication by the cutting equation is injective on every R^jg_*\mathcal F, by the associated-point choice. The induction hypothesis on C and Serre vanishing for the middle terms therefore give H^i(Y,R^jg_*\mathcal F)=0 for i\ge2.

The Leray spectral sequence now has only columns zero and one. For each j it identifies H^1(Y,R^jg_*\mathcal F) with the first filtration subspace of H^{j+1}(V,\mathcal F). After the twist by g^*(mA) this subspace maps to zero, since all positive base cohomology of the twisted direct images vanishes. Functoriality of the Leray filtration and Fujino–Matsumura injectivity imply that the original subspace is zero. This proves the remaining degree i=1.□

Proof. Fix a<b in S. The affine relations

will make the fiberwise section direction unique. Put e=\kappa(X,N_b). If e=0, the two effective divisors in (11) are equal. Letting s tend to infinity gives N_b\ge N_a, so N_b-N_a\sim(b-a)L is effective. If e=\dim X, bL=N_b+F_0 is big, which also proves the assertion. We treat 0<e<\dim X.

Resolve a multiple system of N_b attaining its Iitaka dimension and take its Stein factorization. After modifying the base, flattening the main component, normalizing, and resolving, we have

Here Y,V are smooth projective, W is normal projective and birational to X, f_W is equidimensional, and f has connected fibers. The normalization is finite and preserves the fiber-dimension bound from flattening; the resolution V\to W need not be equidimensional over Y. Equivalently the flat main component is obtained from the closure of the family of generic fibers in the projective Hilbert scheme [16, Theorem 3.2]. Write \pi:V\to X, L'=\pi^*L, and E=K_{V/X}\ge0. For a very ample A on Y and some integer t>0,

The initial moving polarization stays big after base modification, so it dominates the chosen A after increasing t.

For a geometric generic fiber F,

Indeed adding an effective exceptional divisor changes no global sections of multiples of a pullback from the smooth X. If two sections on the generic fiber were independent, a sufficiently large ample base twist would globalize two such sections. Absorb that twist using t\pi^*N_b\ge f^*A, up to linear equivalence. Their nonconstant fiberwise ratio, together with the ratios giving the base, would yield image dimension at least e+1 for a multiple of \pi^*N_b+E. The geometric generic fiber is integral because the Stein field is relatively algebraically closed in characteristic zero. Generic base change justifies the globalization over the original function field. This contradiction proves (12). The same assertion holds without E.

Restriction of (11) to F is therefore equality of effective divisors. Unboundedness of S shows (\pi^*(N_b-N_a))|_F\ge0. If d=b-a, a rational section z of dL with divisor N_b-N_a is consequently regular on F. For every k\ge0,

The indicated effective section gives the lower bound. For k\ge1, multiply by the section of k\pi^*N_a+(k-1)E to inject into the sections of k(\pi^*N_b+E)|_F. For k=0, multiply by the section of \pi^*N_b|_F. Equation (12) gives both upper bounds. This explicitly includes the zero exponent.

On the normal equidimensional model W, take for every prime P\subset Y the minimum

Every vertical prime maps to a base prime and every denominator is positive. There are only finitely many nonzero c_P. Subtracting f_W^*(\sum c_PP) leaves an effective rational Cartier divisor, because the horizontal divisor of z was already effective. Clear denominators, replacing d,z by a common power. We obtain

with H pulled back from an effective Cartier divisor on W and B integral on Y. Above each base prime at least one component has coefficient zero in H.

Proposition 2.2 now applies with the ordinary line P=L, the rational section z, and the normal equidimensional model W. Equation (13) supplies rank one for every k\ge0; the vertical minima giving (14) agree with the proposition’s normalization. Hence B has a continuous semipositive metric. In a local frame b, with \sigma=s_Hf^*b and h_0=\pi^*h_L, its weight is

Only local boundedness of this metric is needed in the following injectivity argument.

It remains to use the zero exponent rather than merely asymptotic positivity. Since F_0 is pseudoeffective, the ample domination above gives nL'-f^*A pseudoeffective for some n>0. Its positive current yields a singular metric on L' with curvature dominating a positive multiple of f^*\omega_A. Mix its weights with the smooth semipositive weights using a sufficiently small positive coefficient. Skoda’s small-exponent integrability [9, Lemma 5.6(a)], in the squared-norm convention e^{-\phi}, and a finite covering of V make the new multiplier ideal trivial while retaining a strictly positive base curvature coefficient. No analytic-singularity assumption is used. For k\ge0 put C_k=L'+kf^*B and add kf^*\phi_B to the coefficient weights. Local boundedness keeps the multiplier ideal trivial and preserves the positive base-curvature lower bound. Since K_V+C_k=E+kf^*B, Lemma 6.1 and the projection formula give

Riemann–Roch makes the Euler characteristic a polynomial in k. At k=0 it equals h^0(V,E)=1. It is therefore nonzero at some positive integer k. A section of E+kf^*B, multiplied by s_H^k, gives a section of E+kdL'. Since \pi_*\mathcal O_V(E)=\mathcal O_X, it yields a nonzero section of kdL on X.□

Proof. For p=0 the assertion is immediate. Otherwise take the stable generic span of the directions of the twisted forms in \Omega_X^p, of rank r>0, and saturate its determinant in \bigwedge^r\Omega_X^p\subset(\Omega_X^1)^{\otimes pr}. A saturated rank-one subsheaf of a locally free sheaf on smooth X is reflexive and hence invertible; call it M. Wedge an arbitrarily far-tail member with suitable r-1 members of a fixed generic basis. This gives nonzero sections of M+sL for unbounded sums s. The cotangent-tensor generic-nefness theorem [18, Theorem 4.1] applies because -K_X is nef and proves that -M is pseudoeffective. Apply the proposition with F_0=-M and the divisors of these sections.□

Proof. This is the regularized-maximum construction of Richberg [28]; see also [10, Theorem I.5.21]. The small negative curvature allowance permits its use for semipositive, rather than strictly positive, weights. Write a weight as a smooth reference weight plus a global continuous function u, with background curvature \gamma. For fixed \varepsilon>0, use background \gamma+\varepsilon\eta, so that the original potential has a strict local margin. On finitely many enlarged coordinate patches, add a smooth local potential of this background, convolve, and subtract that potential. Take inner patches which still cover Y. Add smooth bumps zero on the inner patches and a small negative constant near the outer boundaries, choosing their second derivatives smaller than the strict margin and the convolution errors much smaller than the bump heights. A regularized maximum of the active candidates is smooth: a candidate near its outer boundary is uniformly below a candidate from an inner patch and does not affect the maximum. Convexity, monotonicity, and translation equivariance of regularized maximum preserve the background lower bound and the transition rule for weights. Let the uniform error and then \varepsilon tend to zero. Equivalently, one may smooth the strictly positive metric on NQ+H, with H positive, subtract the weight of H, and divide by N.□

Proof. Fix k. We will show that every nonzero cohomology class on the source restricts nontrivially to some general fiber. Smooth-coefficient decomposition will then give the asserted vanishing. Choose a Kähler form \omega on V and a Dolbeault class c\in H^\ell(V,K_V+A), where A=D_V+kf^*Q and n=\dim V. Lemma 7.1 gives product metrics h_j\to h uniformly on A, with curvature at least \theta-o(1)\omega. Let u_j be the harmonic (n,\ell)-representative of c for h_j. Harmonic norm minimization, comparison with one fixed smooth closed representative, and uniform metric equivalence give a fixed L^2 bound. The Bochner–Kodaira identity gives

Here curvature commutators in bidegree (n,\ell) are monotone in the curvature form: their eigenvalues are sums of the curvature eigenvalues on the antiholomorphic index set.

Let v_j=*u_j, absorbing a fixed nonzero dimensional constant so that u_j=\omega^\ell\wedge v_j. The complex-linear star acts on form indices, carrying the coefficient line along. Metric compatibility gives (D'_j)^*=\pm*\bar\partial* and \bar\partial_j^*=\pm*D'_j*. Thus

These are statements about a fixed holomorphic bundle; the first operator does not differentiate the varying metric. Passing to a weak L^2 subsequence gives a distributionally holomorphic, hence smooth holomorphic, v. Since \omega is Kähler, the form u=\omega^\ell\wedge v is \bar\partial-closed and represents c. Indeed projection to harmonic forms for one fixed smooth reference metric is a bounded finite-rank operator, has the same value on every u_j, and commutes with this weak limit. The nonnegative square root of [\theta,\Lambda] annihilates the limit by (18). Pointwise diagonalization therefore shows that every positive-curvature holomorphic covector is a wedge factor of v.

The approximation also gives a Chern equation before any use of the kernel condition. In any local holomorphic frame of A, let h_j denote its squared norm. The equation D'_jv_j=0 reads \partial(h_jv_j)=0. Uniform convergence of h_j and weak L^2 convergence suffice to pass it to distributions:

No derivative convergence of metric weights is used.

To detect the class on a fiber, we next show that v contains all base canonical covectors as wedge factors. On the open set in (16), every pulled-back base covector annihilates \ker\theta, so its wedge with v vanishes. The holomorphic bundle map

therefore vanishes everywhere. Over the smooth fibration locus, in base coordinates y_1,\ldots,y_b, this gives

The relative restriction w_y is unambiguously defined after choosing the base canonical frame. If n-\ell<b, then v=0 already.

The limiting metric is the smooth metric of D_V times a positive continuous function pulled back from the base. Because of (20), only vertical differentiations survive in (19). The continuous base function is constant under these differentiations; the remaining expression is smooth in the vertical variables and continuous altogether. It vanishes pointwise and gives D'_{D_V|V_y}w_y=0 on every smooth fiber under consideration.

The restriction in cohomology is restriction of a (0,\ell)-form valued in K_V+A. Use K_V|_{V_y}=K_{V_y}\otimes K_Y|_y to remove the constant canonical base factor. It is not the ordinary pullback of an (n,\ell)-form to a lower-dimensional fiber. The restricted representative is a nonzero constant times

It is harmonic on the compact smooth fiber: it is \bar\partial-closed, and its star has zero Chern derivative. Thus a nonzero v gives a nonzero restriction class on some general fiber. The Leray edge map

is injective, since zero edge image implies zero restriction on the general smooth base-change locus.

Finally apply Fujisawa’s derived decomposition [15, Theorem 1], obtained from Takegoshi’s harmonic representatives, to the smooth coefficient D_V:

Tensor the resulting derived isomorphism by the line Q^k on Y and use the projection formula. It gives

Injection (21) bounds the left side by the summand p=0. All remaining summands vanish. For total degrees exceeding n, the left side is zero directly. This proves every case of (17), including the twist k=0.□

Proof. Keep the uniformly convergent metrics and the notation of the proof of Theorem 7.2. For \ell>0, put

with coefficient pairing understood. The distributional Chern equation and holomorphy make T a continuous closed positive (n-1,n-1)-form. The curvature-energy equality gives \theta\wedge T=0. Choose a positive curvature form \eta of H. A positive current in the class of bD_V-f^*H pairs nonnegatively with T, since positive currents have order zero. To compute this pairing cohomologically, apply de Rham regularization, which commutes with d, to approximate T uniformly by smooth closed forms. For each approximant the pairing agrees with that against the cohomologous smooth form b\theta-f^*\eta; uniform convergence passes this equality to T. It follows that

Pointwise positivity forces f^*\eta\wedge T=0; diagonalization gives the full base wedge factor in v. The distributional fiber equation, canonical-valued restriction, edge injection, and smooth-coefficient splitting proceed exactly as in that proof. Only positive total degrees are needed to kill a term of positive base degree.□

Proof. Let \mathcal G=f_*\mathcal O_V(E), or its invariant summand in the equivariant case. A torus over a trivially acted-on base has coherent weight summands, so this is a direct summand. Vanishing gives h^0(Y,\mathcal G\otimes Q^k)=\chi(Y,\mathcal G\otimes Q^k) for every k\ge0. The right side is polynomial in k by Riemann–Roch for coherent sheaves. At zero it is one: \pi_*\mathcal O_V(E)=\mathcal O_X and the exceptional canonical section is invariant. The polynomial is therefore nonzero at some positive integer. Multiply the resulting section of E+kf^*Q by the k-th power of the injection and push through \pi. The effective exceptional divisor E changes neither sections nor their characters, by normality of X.□

Harmonic proof of Corollary 4.3. Use Lemma 4.2, retaining the cases which did not already produce a section, and write N_i=M+a_iL, s_i for its invariant sections. Write n=\dim X, let \theta be the averaged smooth semipositive curvature, let r be its maximal rank, and fix a Kähler form \omega. Choose indices a_-<a and keep unbounded indices above a. Put N=M+aL. Among rational maps given by all invariant sections of positive powers of N, choose one with maximal image dimension. Resolve its base ideal equivariantly and take Stein factorization. We obtain a smooth projective X_0, a birational \pi_0:X_0\to X, and f_0:X_0\to Y_0 with connected fibers and normal projective base, with

where H_0 is ample. The torus acts trivially on Y_0,H_0, and the defining section of G_0 is invariant.

The invariant sections of j\pi_0^*N on the generic fiber form a one-dimensional vector space over \mathbb C(Y_0) for every j>0. Indeed finitely many generic-fiber sections extend rationally with vertical poles. A section of a sufficiently high power of H_0 clears those poles; multiplication by the corresponding power of the section of G_0 gives invariant sections of a higher power of \pi_0^*N, and hence of N on X. If two original sections were independent, their ratio would not lie in \mathbb C(Y_0). This field is relatively algebraically closed by Stein factorization, so that ratio would be transcendental over it. Combine it with the original image coordinates, by products to put all ratios in one power. The image dimension increases, contradicting maximality.

For each a_i>a there are positive integers p,q such that pa_i+qa_-=(p+q)a. The invariant section s_i^p s_-^q is thus generically proportional to s_a^{p+q}. Its effective generic-fiber divisor shows that all horizontal components of \operatorname{div}(\pi_0^*s_i) lie in the fixed finite support of \operatorname{div}(\pi_0^*s_a). Extract a subsequence of their integer multiplicity vectors that is componentwise nondecreasing. For two indices a_j>a_i in it, put

This is an invariant rational section of b\pi_0^*L, regular on the generic fiber. The relative canonical divisor E_0=K_{X_0/X} has its horizontal support in the same fixed support: \pi_0 can be chosen to be an isomorphism outside the invariant base ideal, which lies in the zero locus of s_a. Fix k\ge0, and work on the generic fiber. The effective divisors of s_{E_0} and s have support in the divisor of \pi_0^*s_a. All coefficients are finite, so a sufficiently large integer h satisfies

The quotient (\pi_0^*s_a)^h/(s_{E_0}s^k) is therefore a regular invariant section of the difference line. Multiplication by it injects the entire invariant adjoint section space into the one-dimensional invariant space for h\pi_0^*N. Consequently

If \dim Y_0=0, the section s is already regular on X_0 and descends to the required section on X.

Otherwise apply Lemma 2.1 to this fibration and resolve its normal equidimensional model:

Write \mu:W\to X and \pi:V\to X for the composed birational maps. We also use f_0 for its pullback to V. Here Y is smooth projective, W is normal equidimensional over Y, and V is smooth. All constructions are equivariant, with trivial base action. The ranks (23) persist: a further effective birational canonical divisor has trivial pushforward also on the generic smooth fiber.

It remains to relate this algebraic fibration to the curvature. The kernel condition for D_V=\pi^*L follows from Lemma 4.4 and (22). If r<n,

Indeed the whole class dN has zero intersection, and both the moving and effective fixed parts have nonnegative intersection. On an ordinary open where \pi is an isomorphism, \theta has rank r, and the base-to-image map has injective differential, positivity forces the horizontal form to annihilate \ker\pi^*\theta. Its kernel there is \ker df. Thus \ker\pi^*\theta\subset\ker df. These algebraic generic conditions meet the nonempty ordinary maximal-rank open. For full curvature rank the kernel inclusion holds directly.

Proposition 2.2 normalizes s to g^*Q\to b\mu^*L, with effective zero divisor G, and gives a continuous semipositive metric on Q. Theorem 7.2 and Proposition 7.5 now produce an invariant section of a positive power of L.□

Proof. On V, form

viewed as sheaves of rational functions on Y by division by s_Es_G^k. They have rank one, contain \mathcal O_Y, and increase with k. For each prime P of Y, take a lift \Gamma omitted by G. Regularity of f^*a\,s_Es_G^k gives

The possible poles are therefore bounded independently of k by one divisor C. Normality embeds all \mathcal P_k into \mathcal O_Y(C), so Noetherianity gives a stable value \mathcal P_\infty.

Take a very ample H on Y. Apply Theorem 7.2, with trivial base twist, to the smooth coefficient (1+kb)D_V+lf^*H, k,l\ge0. Its curvature dominates the original curvature, so its kernel is still vertical on the same open set. Since its adjoint line is E+kG+f^*(kQ+lH), projection to invariants and the projection formula give

For large k, Castelnuovo–Mumford regularity makes \mathcal P_\infty\otimes(kQ+(\dim Y)H) globally generated. Pull back to the normalization of any curve and remove torsion. The fixed sheaf has positive rank, so its degree plus k times that rank times \deg Q is nonnegative for unbounded k. Thus Q is nef.

If G contained the entire fiber over y\in Y, blow up y (the curve case is already excluded by normalization). The normalized main transform remains equidimensional because it lies in a base change with the same fiber-dimension upper bound. Recompute the vertical minima on this model. The new rational base divisor is the pullback of Q plus a strictly positive multiple of the exceptional divisor: every divisorial lift of its generic point lies over the old fiber and therefore has positive order in the pulled-back G. The rank and kernel arguments just given still apply, so the new divisor is nef after clearing denominators. Its degree on a line in the exceptional projective space is negative, a contradiction. Hence no entire fiber lies in G. Properness and openness give a positive continuous maximum, and the moment calculation (2) proves semipositivity without the codimension-two extension argument. This gives the same conclusion by a distinct pole-sheaf stabilization proof.□

Proof. An ample line has a linearized positive power for a projective \C^*-action. Indeed the action’s map to the Picard variety is constant, since a torus has no nonconstant homomorphism to an abelian variety. The pairs consisting of an element and an isomorphism from the pulled-back line to the line form an algebraic extension of \C^* by the scalar \C^*; this extension is a torus and splits. This supplies the needed linearization. The section-weight bound, the comparison of fixed weights, and the index formula are Lemma 5.1 for T=\C^* and U as given. The source retraction itself is the attracting-cell map of [7]. The additional assertion needed by the harmonic construction is its bundle map, which we now prove.

For the bundle map, identify the generic fibers of U and its restriction at the source by transporting a vector along an orbit, dividing by t^{w_{S_0}(U)}, and taking the limit. The orbit extension over \mathbb A^1, over the relevant function field, pulls the line back to an equivariantly trivial line of that weight. This constructs a multiplicative rational identification, hence the claimed rational map. If U is ample, a generated power has enough source-weight sections to generate its restriction to S_0. Their pulled-back restrictions map to the original sections by the limit construction, proving absence of poles. For nef U, a negative order in the rational map would give a negative order for kU+H for large k, contrary to ampleness. This proves regularity along every prime divisor and hence everywhere by normality. The trivial action has S_0=S and the identity map.□

Proof. Determinant sections with a prescribed weight. A point is immediate. Otherwise S_0 is rationally connected, being a smooth rational image of S. Thus \chi(S_0,\mathcal O_{S_0})=1. The highest-weight index in Lemma 10.1 is a nonzero polynomial in m, so for unbounded m a fixed group H^q(S,mL) contains weight mw. The smooth coefficient Lefschetz map is equivariant for a circle-invariant Kähler form. It gives twisted sections

of weight mw. For q=0 they are the desired sections. Otherwise saturate their generic span in \bigwedge^qT_S, of rank e>0, and call it \mathcal U. The determinant extraction used earlier gives nonzero eigensections

If D_i is the zero divisor, we claim

For q=\dim S, saturation gives \mathcal U=\bigwedge^{\dim S}T_S=L and e=1, so eL-c_1(\mathcal U)=0 directly. For 0<q<\dim S, the cotangent-subsheaf theorem [18, Theorem 4.1] applies: \mathcal U\otimes(-L)\subset\Omega_S^{\dim S-q}, and exterior powers are direct summands of tensor powers in characteristic zero; saturating can only add an effective determinant correction in the required direction.

A base on which dominated systems become rigid. Call a divisor class dominated if a positive multiple of L minus it is pseudoeffective. Maximize the image dimension of linear systems of dominated effective integral divisors, and choose a full system |P_0| realizing this maximum. Rational connectedness gives \operatorname{Pic}^0(S)=0, so the system may be linearized and taken equivariantly. Resolve its graph, take Stein factorization, flatten by the Hilbert-family construction, resolve the base and normalize the main component. We obtain

with smooth projective V,Y, normal equidimensional W, and connected fibers. Write f:V\to Y, \mu:V\to S, A=\mu^*L, E=K_{V/S}, and let L_W be the pullback of L to W. An ample pullback from Y is dominated: the original moving polarization is dominated, and its pullback to a birational base model is big, hence dominates a small ample class. Also E is dominated. Horizontal exceptional primes have centers in the original base locus and are bounded by pullbacks of divisors in |P_0|; vertical primes are bounded by sufficiently ample base pullbacks. These statements prove domination of the whole finite effective canonical discrepancy.

A line B on V is dominated when cA-B is pseudoeffective for some c>0. No such line can have two independent sections on the generic fiber. If it did, tensor by a sufficiently ample base pullback to globalize them. This new line remains dominated because the base polarization is dominated. Combine their ratio with the original base coordinates, using products to put all coordinates in a single linear system. Its line is still dominated. Push the effective system and its pseudoeffective domination relation down to S. The new ratio is nonconstant along the generic fiber, so the image dimension exceeds the chosen maximum, a contradiction. In particular generic-fiber sections of both jaA and E+jaA have dimension at most one for every a,j>0, and the same adjoint assertion for j=0 follows from domination of E.

There are only finitely many dominated horizontal primes on V. Here \operatorname{Pic}^0(V)=0, since V is birational to the rationally connected S. Thus its Picard group has finite rank. Infinitely many dominated horizontal primes would give, after clearing torsion, a nontrivial integral linear equivalence among them. Splitting positive and negative coefficients gives two effective divisors in the same class and a pencil varying along the generic fiber, contrary to the preceding paragraph. The pullbacks of D_i therefore have horizontal supports in a fixed finite set. Order their multiplicities along a subsequence. A ratio from (25) gives a rational eigensection s of aL_W, weight aw, with no horizontal poles.

The metrized base line and its source weight. Divisorial normalization, as in Proposition 2.2, gives

with effective zero divisor omitting one lift over every base prime. Shift the linearization of Q to make this map equivariant. Since the canonical exceptional section times s^j is nonzero generically, all required adjoint ranks are exactly one. The maximum construction gives a continuous semipositive metric on Q. Here the base action may be nontrivial, but full rank one is available and the ordinary direct-image line theorem applies without taking an invariant summand.

We must still control the character produced by interpolation on the base. Let Y_0 be the source of its induced action. The original unbounded eigensections imply

After fixing a, take an infinite subsequence of k_i in one residue class modulo a, still ordered horizontally. Divide by its first section. The resulting eigensections of jaL_W, for unbounded j, have weight jaw, no horizontal poles and vertical poles bounded by one fixed divisor. Clear these poles by an invariant effective Cartier divisor C on Y, with its canonical section of weight zero. Generic rank one and the map (26) show that each resulting section comes from jQ+C. It has no residual poles: at a base prime test on a lift omitted by the zero divisor of (26). The source upper bound gives jaw\le j w_{Y_0}(Q)+w_{Y_0}(C). Divide by j and pass to the unbounded subsequence to obtain (27).

Euler interpolation on the source of the base. The continuous semipositive metric makes Q nef by Lemma 7.1. Resolve the source retraction of Y and dominate this resolution by a smooth model V' of V. Let p:V'\to Y_0, \mu':V'\to S, A'=\mu'^*L, E'=K_{V'/S}, and Q_0=Q|_{Y_0}. Lemma 10.1 and (26) give p^*Q_0\to aA'. An ample pullback by p is dominated by A': the pullback of the original ample base polarization is big on the resolution of Y, and hence dominates an ample class there, which in turn dominates an ample pullback from Y_0. Proposition 7.4 applies to every twist jQ_0, including zero. The action on Y_0 is trivial, and Q_0 has the constant fiber weight w_{Y_0}(Q). Let \mathcal B be the weight-zero summand of p_*\mathcal O(E'). Thus sections of \mathcal B\otimes Q_0^j have precisely weight j w_{Y_0}(Q). Vanishing identifies their dimension with the Euler polynomial for every j\ge0. At zero this polynomial is positive, because the canonical exceptional section is invariant. For some j>0 we therefore obtain a nonzero section of E'+jp^*Q_0 of that weight. Map it into E'+jaA' and push down to S. Its weight is at least jaw by (27), and at most jaw by the source bound on S. Equality gives the required extremal section.□

Proof. For a generic integral one-parameter subgroup \lambda, its source C is a torus-fixed component with all normal weights positive on \lambda. Denote by \gamma_C the fiber character of L. Choose a full torus eigenvector in the extremal section space of Proposition 10.2. It cannot vanish on C, since a nonzero leading conormal jet would strictly lower its \lambda-weight. Its character is therefore \ell\gamma_C. The finitely many characters of such sources have convex hull containing zero. Otherwise a separating functional, approximated by a generic rational one, would be negative on all of them, whereas the source for that one-parameter subgroup has \gamma_C(\lambda)>0, the sum of its positive normal weights. Use the effective torus quotient if the action has a kernel; a trivial action is already covered. A rational convex combination of the source characters is zero. Clear its denominators and the finitely many section exponents, then multiply the corresponding powers of their nonzero eigensections. On the integral variety this product is nonzero and invariant.□

Proof. We construct a globally strictly positive metric with exactly the same multiplier ideal.

Producing an integrable divisor metric. Choose a very ample A with curvature \omega_A, and a smooth metric \psi_0 on R. For sufficiently small fixed \delta>0 and arbitrarily large integers l, we first claim that

On lR use weights l(1+\delta)\psi-l\delta\psi_0, and add the smoothly metrized line j_lA-K_Y, with j_l=\lceil C\delta l+C'\rceil. Fixed sufficiently large constants C,C' absorb the negative smooth curvatures and leave a positive lower bound. The multiplier ideal is \mathcal I_l=\mathcal I(l(1+\delta)\psi). Within a smaller ball of strict positivity choose a point and a cutoff logarithmic pole equal to \log|z|^2 near that point. Add \eta l times this pole, with \eta>0 fixed sufficiently small independently of \delta,l. Its negative curvature costs at most O(\eta l) on the ball, where the original positive lower bound has order l. Nadel vanishing [23], in the strict-current form of [9, Theorem 5.11], therefore gives surjectivity to jets of order \lfloor\eta l\rfloor-d at the point. Continuity of \psi there makes \mathcal I_l trivial near that point, and polar-coordinate integration gives the ideal \mathfrak m^{\lfloor\eta l\rfloor-d+1} for the added pole. Consequently

for some c>0 independent of sufficiently small \delta.

Remove j_l+1 copies of A. At each step choose a smooth member of |A| containing no associated center of \mathcal O_Y/\mathcal I_l. If a section vanishes on that member, division by its equation preserves membership in \mathcal I_l, because that equation is a nonzerodivisor on the quotient. The number of conditions lost at a step is at most C''l^{d-1} uniformly for 0<\delta\le1. Indeed choose p with pA-R globally generated, restrict a general nonzero section to the hyperplane, and bound the restriction space by that of (pl+h)A there, where 0\le h\le j_l. The Hilbert polynomial and the hyperplane exact sequence give this bound. The total loss is O(\delta l^d)+O(l^{d-1}). Choosing \delta small and then l large proves (32).

Let \sigma be a nonzero section in (32). In compatible local frames, the weight \phi_A of the positive metric on A defines a divisor metric on R by

Its curvature dominates \omega_A/l. If V=\psi'-(1+\delta)\psi, membership of \sigma in the multiplier ideal in (32) says exactly that \exp(lV) is locally integrable.

Preserving the ideal. Take l\ge1/\delta and put \rho=(1-\varepsilon)\psi+\varepsilon\psi'. The identity

and Hölder’s inequality, with the integrability of \exp(V/\delta), prove \mathcal I(\rho)\subseteq\mathcal I(\psi). For the converse, strong openness [17, Theorem 1.1], coherence, and compactness give \gamma>0 with \mathcal I((1+\gamma)\psi)=\mathcal I(\psi) on a finite cover. Small-exponent integrability of the divisor metric \psi' gives \gamma'>0 with \exp(-\gamma'\psi') locally integrable. Apply Hölder with s=(1-\varepsilon)/(1+\gamma) and 1-s. For small \varepsilon one has \varepsilon/(1-s)<\gamma', proving \mathcal I(\psi)\subseteq\mathcal I(\rho). The metric \rho has a global positive curvature lower bound and the same ideal. Nadel vanishing proves the assertion.□

Proof. We spell out the consequence of Berndtsson’s curvature formula [5, Section 4, Proposition 4.2, Lemmas 4.3–4.4, and equation (4.8)], since strict vertical positivity is not required. Restrict to a disk with coordinate z in a chosen base direction, and let the fiber dimension be a. If a=0, the direct image is locally the orthogonal sum of the coefficient lines on the finitely many sheets. Its curvature pairing is a sum of nonnegative terms, with a positive term on the patch where u_0 is nonzero. Assume now a>0. A holomorphic direct-image section extending u_0 with vanishing covariant derivative at zero has a smooth absolute (a,0) representative u satisfying

On the central fiber, \lambda is orthogonal to holomorphic adjoint forms and \eta\wedge\omega is \bar\partial-exact. Replace u by u-dz\wedge\xi so that \lambda=0 and \eta is primitive there. These two requirements prescribe, respectively, \bar\partial^*(\omega\wedge\xi) and \bar\partial(\omega\wedge\xi), up to the Kähler-identity constants. Orthogonality and exactness place the two prescriptions in their required images; Hodge decomposition solves them simultaneously. Lefschetz linear algebra identifies the relevant (a,1)-forms with \omega\wedge\xi, for \xi of type (a-1,0).

With c_a=i^{a^2}, the resulting curvature pairing is

The second term is nonnegative by the primitive middle-degree Hodge–Riemann sign. The first integrand is nonnegative everywhere. On the prescribed patch the pullback lower bound contributes the positive base form times the squared vertical norm of u_0. This is positive on a nonempty open subset, so its integral is strictly positive. For a torus action, averaging is a holomorphic orthogonal projection. The Chern connection preserves the orthogonal character splitting, so the same calculation applies on the invariant summand.□

Proof. The sheaf \mathcal B is torsion-free. Its hull is determined by regularity over the generic points of base primes. More precisely, for a prime P\subset Y its coefficient in B_0 is

These are exactly the regularity tests for the rational section (f^*a)t. On the smooth base-change open, the entire invariant adjoint summand is a line. Its fiber-integral metric is semipositive by the compact-torus orthogonal quotient argument of Proposition 2.2.

Put k=\dim Y. This metric extends plurisubharmonically across every base prime. At a general point of a component attaining b_P, write the map as (z_1^e,z_2,\ldots,z_k), where e>0 is the pullback multiplicity, and let a be the order of t. The relative canonical coefficient for the local frame of R is a nonvanishing smooth factor times

Since 0\le a-b_Pe<e, this exponent is nonpositive. Integration on one fixed small vertical polydisk bounds the squared norm below by a positive constant near the base prime. The weight is therefore locally bounded above there. Extend first along these divisorial neighborhoods and then across codimension at least two; write the extended weight as \psi_R.

Choose local frames g of \mathcal O_Y(B_0) and \eta of K_Y. The frame g/\eta of R corresponds on the smooth locus to the relative adjoint section (f^*g)t/f^*\eta. For a holomorphic function a, Fubini identifies integrability of a with this frame’s squared norm, using the base volume from \eta, with square integrability of the absolute top form (f^*(ag))t with the smooth L' metric upstairs. The excluded analytic sets have measure zero. Such an integrable meromorphic top form has no divisorial pole; smoothness upstairs then extends it across codimension two. Conversely a holomorphic lift is integrable over inverse images of relatively compact base charts by properness. The lift is invariant and the identification agrees on the generic fiber. This proves both inclusions in (33), so it retains any codimension-two difference between \mathcal B and its hull.

For the final assertion apply Lemma 12.2 to the invariant section determined by t on the stated patch. The fiber-integral metric is smooth on the base-change open, so its strictness at one point gives a positive lower curvature bound on a smaller coordinate ball.□

Proof. The null-pole field and the relative section. Average a metric on L and a Kähler form \omega over the maximal compact torus, and write \alpha\ge0 for the resulting curvature form, in first-Chern normalization. Proposition 4.1, with its actual value at zero, followed by the equivariant hard Lefschetz construction in Lemma 4.2, has the following consequence: if the asserted invariant sections do not exist, there is a fixed q>0 and unbounded positive j with

The coefficient in hard Lefschetz is j=m+1, since K_X+jL=mL equivariantly. In the Bochner identity for the associated harmonic (n,q)-forms the curvature term is a sum of q eigenvalues of j\alpha. Full rank on an open set would force vanishing there, hence vanishing of the holomorphic Lefschetz preimage everywhere. Thus r=\max\operatorname{rank}\alpha<n.

Define the degree of a divisor by

It is nonnegative on effective divisors and vanishes on L. Lemma 4.2 gives a linearized determinant line S and, for unbounded positive integers p, nonzero invariant sections

Put N_p=\operatorname{div}(s_p). Lemma 4.4 gives \ell(N_p)=0: prescribed Ricci curvature first gives \ell(S)\le0, and effectivity supplies the reverse inequality. The degree-zero divisors are exactly the curvature-null divisors of Section 9.1. We use their invariant pole field again, now to construct the exact direct-image ideal.

Let K be the field of invariant rational functions whose polar divisors have degree zero. Sums and products preserve this property; inverses do also, because polar and zero divisors are linearly equivalent and effective degrees are nonnegative. The field is relatively algebraically closed in \mathbb C(X). Indeed a root of a monic equation over K can have poles only on the polar divisors of its coefficients, by the discrete-valuation inequality. The connected torus fixes each member of the finite root set, so the root is invariant as well. The field is finitely generated: choose a transcendence basis over \mathbb C; algebraic subextensions over its rational field inside the finitely generated ambient field have bounded degrees, after adjoining an ambient separating transcendence basis. Their compositum is therefore finite.

Take a smooth projective model Y of K. Flatten the graph of the induced rational map, and normalize its main component, to obtain a normal equidimensional G\to Y birational to X; then take an equivariant resolution V\to G. The action on Y is trivial. Write f_G:G\to Y for the normal-model morphism and L_G for the pullback of L to G. Write \pi:V\to X, f:V\to Y, L'=\pi^*L, and let t be the invariant Jacobian section of K_V+L'=\mathcal O_V(J), with J=K_{V/X}\ge0. The generic fiber is geometrically integral because K is relatively algebraically closed. These constructions use only equivariant resolution of the graph and the Hilbert-family flattening; they do not assert equidimensionality of the smooth resolution.

Ratios of products of the s_p having the same bidegree in (S,L) belong to K. Their horizontal divisor on G is zero, so for p>p_1>p_0,

Equivalently,

Effectivity for unbounded p forces N_{p_1}^{\rm hor}-N_{p_0}^{\rm hor}\ge0. Put d_0=p_1-p_0>0 and \sigma_0=s_{p_1}/s_{p_0}, a rational invariant section of d_0L with effective horizontal divisor. We use the same symbol for its pullback to G. If Y is a point, it has no poles and proves the assertion. Assume \dim Y=k>0.

For each prime P\subset Y, put

Only finitely many c_P are nonzero. Choose a positive integer h clearing their denominators, and set

Thus \sigma is a rational invariant section of dL_G, and B is an integral divisor with trivial action. Divisorial normalization gives

Here 1_B is the rational canonical section of \mathcal O_Y(B). At least one component above every base prime has coefficient zero in D. We also write \sigma for its pullback to V.

Adjoint rigidity and a horizontal curvature bound. For every m\ge0,

There is a generic section t\sigma^m. If the invariant rank were larger, a large very ample base twist would give two invariant global sections independent over K. Their pushed divisors on X have degree zero. For the base twist this follows by taking the ratio of two general base hyperplanes: their pullbacks to the equidimensional G have no common prime component, so their pushed divisors are the actual polar and zero divisors of an element of K. The remaining class is mdL, and the exceptional Jacobian pushes to zero. Thus the ratio of the two sections would again belong to K, contradicting their independence. This proves (35), including m=0.

At a rank-r point of \alpha, the differential of every rational function in K kills \ker\alpha wherever it is regular. For a regular value, the closure of its level hypersurface has degree zero. The nonnegative form \alpha^r\wedge\omega^{n-r-1} therefore vanishes on that hypersurface. A tangent hyperplane failing to contain \ker\alpha would still carry rank r, a contradiction. Apply this to k independent base coordinates. It gives k\le r. Intersect the maximal-rank open with the dense smooth base-change locus and the complement of the exceptional locus. On a small patch there,

This is semipositive linear algebra on a complement of the kernel, not an inference from numerical degree alone. If r=0, a nonzero effective divisor cannot have degree zero, so k>0 cannot occur.

Apply Proposition 2.2 to the equivariant normal equidimensional model G\to Y, with P=L naturally linearized and the rational section \sigma. Equation (35) verifies the invariant adjoint ranks, and (34) is its vertical normalization. The result is a positive continuous semipositive metric on B.

The exact ideal and interpolation from zero. The varying twist is now metrized. The degree-zero metric must retain the entire direct image, including its codimension-two defect.

Apply Proposition 12.3 to \mathcal B=(f_*\mathcal O_V(J))^T. Equation (35) at m=0 gives its rank, and the patch in (36) is disjoint from the exceptional locus, so t is nonzero there. The proposition supplies the metric \psi_R on R=B_0-K_Y, strictly positive on a base ball, with the exact identity (33).

Adding m times the continuous bounded metric on B preserves the ideal and the strictness on the ball. Lemma 12.1 therefore gives

At zero, canonical pushforward gives h^0(Y,\mathcal B)=h^0(V,\mathcal O_V(J))^T=1. Thus the Euler polynomial has value one at zero. For some m>0 lift a nonzero section and multiply by s_D^m. The result is an invariant section of J+mdL'. Remove the exceptional Jacobian divisor by pushforward to the smooth X. It gives the required natural invariant section of mdL.□

Proof. Generic adjoint rigidity. Write D(j)=M+jL and fix j_0<j_1 in J. Choose k>0 so that the invariant system H^0(X,kD(j_1))^T has maximal image dimension among these systems for all positive degrees. No finite-generation assertion is needed to attain this maximum, an integer between zero and \dim X. An equivariant principalization of its base ideal gives a smooth projective \pi_0:W_0\to X, isomorphic off the base locus, and a connected-fiber Stein map f_0:W_0\to Y_0 such that

where H is ample on the normal projective Y_0 and G is the effective fixed divisor. The base action and the fiber action on H are trivial, while the defining section of G is invariant. These identifications follow by dividing the invariant generating sections by the common fixed factor. The action on the finite Stein factor is trivial by connectedness of T.

Let K=\mathbb C(Y_0) and let F be the generic fiber. It is normal and integral, with H^0(F,\mathcal O_F)=K. The field K is relatively algebraically closed in K(F): an algebraic element has no poles on proper normal F, as follows from a monic equation over K. For every h>0,

There is the section \pi_0^*\sigma_{j_1}^h. If there were two independent invariants, their ratio would be transcendental over K. They extend as invariant rational sections on W_0 with only vertical poles. A common pullback of a section of a sufficiently large multiple lH clears these poles: prescribe vanishing on their finitely many proper base images. Multiplying also by s_G^l produces invariant sections of a multiple of D(j_1) on X. Products with the original system retain its image field and add their new ratio, contradicting maximal image dimension. This proves (37).

Let D_i^F be the effective divisor of \sigma_i|_F. The invariant products of the same linearized degree are proportional, so for c>j_1 in J,

Thus D_{j_1}^F-D_{j_0}^F\ge0. Put d=j_1-j_0>0 and z=\sigma_{j_1}/\sigma_{j_0}, a rational invariant section of dL. It is regular on F, with zero support in D_{j_1}^F. If the base is a point we already have a regular invariant section on W_0, hence on X.

Otherwise let e_0 be the Jacobian section of K_{W_0}+\pi_0^*L. Its zeros lie over the principalized base locus and, on F, in the support of D_{j_1}^F. For every fixed s\ge0, choose h large enough that, on F,

Such an h exists because both effective divisors on the right have support in D_{j_1}^F. The invariant quotient (\sigma_{j_1}|_F)^h/(e_0z^s|_F) is regular. Multiplication by this nonzero quotient injects the entire invariant adjoint section space into the one-dimensional space in (37). Consequently

Actual lines and affine boundary orders. The generic rank computation will now be upgraded to line bundles whose local frames lift to adjoint sections. After further birational modifications, the same adjoint ranks are preserved by canonical pushforward on a common smooth resolution. No support claim about new exceptional primes is needed. Now apply birational weak toroidalization [1, Theorem 1.1], followed by equidimensionalization by toroidal subdivisions [2, Proposition 4.4]. This gives a birational modification Y\to Y_0 with Y smooth, a normal toroidal Z birational over W_0, and an equidimensional g:Z\to Y. These are birational modifications; no alteration or assertion of reduced fibers is being used. Take a smooth resolution W\to Z and write f:W\to Y, \pi:W\to X, and \pi_Z:Z\to X.

Toroidal singularities are rational. Therefore, for

the sheaf \mathscr E_s\otimes\mathcal O_Y(K_Y) equals

It is reflexive: deleting a codimension-two subset of Y deletes only codimension at least two from Z, by equidimensionality, and the reflexive canonical sheaf on Z extends its sections. The pushforward is torsion-free as well, since the source is integral and the map is dominant.

The toroidal modifications may fail to be equivariant. Nevertheless \mathscr E_s has a T-linearization. Compute the same direct image on an equivariant smooth resolution W' of the main component of W_0\times_{Y_0}Y. A common smooth resolution of W and W' identifies their canonical direct images, since all coefficient twists are pulled back from X and a birational morphism of smooth varieties pushes its canonical sheaf to the canonical sheaf. The identifications are the identifications of rational forms, so preserve the generic torus action. The invariant part

is now an actual reflexive direct summand of rank one by (38); on smooth Y it is a line bundle. Birational canonical pushforward gives

The last equality uses the natural invariant trivialization of K_X+L.

We compute these actual lines. On the regular locus of Z let e_Z be the Jacobian section with effective divisor E_Z. Use the rational generator e_Zz^s for K_Y+A_s. For each prime B\subset Y, let Q_i run over primes of Z mapping onto B, and let

A rational base multiplier a is regular above the generic point of B precisely when b_i\operatorname{ord}_B(a)+e_i+sc_i\ge0 for all i. There are no horizontal poles by (38). The exact allowed pole order is

Only finitely many primes occur with a nonzero slope or intercept. Replace d,z by a common positive multiple and power so that all v_B=\min_i(c_i/b_i) are integers, and write V=\sum_Bv_BB. For all sufficiently large integer s,

Indeed only the indices of minimal slope can minimize for large s. Among them the least intercept, after taking its floor, is at least the unrestricted least intercept at zero. This proves the stated inequality coefficient by coefficient. Set A_\infty=\mathcal O_Y(C_\infty)-K_Y. Then A_s=A_\infty+sV eventually, and

for those s. The eventual family has intercept A_\infty, but this inclusion compares it with the actual zero-degree line A_0. It is this inclusion that will transfer sections produced by interpolation from exponent zero.

Integrable metrics and interpolation from zero. The original moving part, together with -M pseudoeffective, shows that \pi^*L-\epsilon f^*H_1 is pseudoeffective for some ample H_1 on Y and \epsilon>0. The pullback of H to Y is big, so it still dominates a small ample class. Mix a metric representing this pseudoeffectivity with the smooth semipositive metric on \pi^*L, using a small positive coefficient. This gives at s=0 a singular metric with trivial multiplier ideal and curvature at least \delta f^*\omega_Y, with \delta>0. For s>0 use the smooth semipositive metrics on (1+sd)\pi^*L.

The singular adjoint metric and its extension [27, Theorems 3.3.4–3.3.5] give semipositive metrics on \mathscr E_s on its locally free locus. The generic multiplier-ideal inclusion is an isomorphism since the upstairs ideal is trivial. Give A_s the quotient metric for the holomorphic projection onto this summand; quotient positivity is [27, Lemma 2.3.4(2)]. Write its weights as \psi_s. Their description is the logarithm of the squared norm of the dual line frame in \mathscr E_s^\vee. They extend plurisubharmonically across the remaining codimension-two subset of smooth Y, since A_s is a line bundle. They are not identically -\infty, by the nondegenerate finite fiber metric at almost every smooth base-change parameter. An invariant choice of upstairs metric is unnecessary for this quotient argument.

Crucially,

A local frame of the actual summand lifts to a local section of \mathscr E_s. Its fiberwise squared norm bounds the quotient squared norm from above. The lift is a holomorphic adjoint section upstairs, after adding a base canonical frame. Its absolute density is integrable over inverse images of relatively compact charts by the trivial upstairs ideal and properness. Fubini on the smooth locus identifies this integral with the fiber norms integrated against the base canonical volume, proving (42). A frame of a reflexive hull without such a lift would not suffice.

The weight \psi_0 has a strictly positive lower curvature bound on all of Y. To check this across the boundary, choose a smooth function p on Y whose Hessian dominates \omega_Y near a given point and is bounded above by C\omega_Y globally. Multiply the upstairs squared metric by e^{t f^*p} for small t>0. It remains semipositive and has the same ideal. Its direct-image quotient weight is \psi_0-tp almost everywhere, so it is plurisubharmonic everywhere by the extension theorem. A finite covering gives i\partial\bar\partial\psi_0 \ge\delta'\omega_Y for some \delta'>0.

We now put strictly positive metrics with trivial multiplier ideal on A_0+sV for every s\ge0. These are the lines on the left of (41), so vanishing for them will retain the known section count at zero. Fix an integer s>0. Take a sufficiently large integer u in the range of (40), and choose a smooth metric \psi_\infty on A_\infty. In compatible frames the formula

defines a metric on A_0+sV. Its curvature is strictly positive once s/u is small enough. Strong openness [17, Theorem 1.1] and compactness give \eta>0 such that e^{-(1+\eta)\psi_0} is locally integrable on a finite cover. Hölder, together with (42) for this u, proves e^{-\rho_s} integrable if

The smooth \psi_\infty contributes only bounded factors. No exponent improvement uniform in u is needed. At s=0 the original \psi_0 already has these properties.

Nadel vanishing consequently gives

The Euler polynomial in s has value one at zero by (39). Thus it is nonzero for all sufficiently large integers s. Using (41) and then (39) gives a nonzero invariant section of sdL on X.□

Proof of Corollary 1.2. On F, the compact torus preserves a very ample line because it preserves its Chern class and \operatorname{Pic}^0(F)=0. Its projective action has algebraic-torus Zariski closure: a compact connected abelian subgroup of a projective linear group is conjugate into a diagonal compact torus. Passing through the finite special-linear cover, if necessary, provides a linearization of a very ample power. Invariants of the natural anticanonical line for the compact group and its Zariski closure agree. Corollary 4.3 therefore gives a nonzero invariant section of -mK_F for some m>0.

Tensor its pullback to \widetilde X with the m-th powers of the inverse canonical frames on \C^c and V. This nonzero product is invariant under the selected deck subgroup and descends to a section on X'. If F is a point, take its section to be 1 and m=1.

The finite étale norm [24, Lemma (Finite étale norm)] sends this section to a nonzero section of a positive power of -K_X. Explicitly, pull to a connected finite Galois refinement, multiply all Galois translates, and descend their invariant product. The product is nonzero on the integral cover, and canonical bundles pull back under étale morphisms. Projectivity and GAGA make the resulting holomorphic section algebraic.□

Proof. For a holomorphic covariant tensor, the contracted curvature on its bundle is the sum, over its tensor slots, of the negative Ricci operator. It is nonpositive. Integration of the Bochner identity on compact P therefore makes the tensor parallel. Its value at a point must consequently be fixed by the holonomy representation.

Let H be the closure of the holonomy group on the complex tangent space at that point. It is a compact subgroup of the unitary group. Simple connectivity makes full holonomy equal to restricted holonomy, so H is connected. De Rham irreducibility makes its complex tangent representation irreducible: a proper invariant complex subspace would give a proper real parallel subspace and its orthogonal complement. Consider the determinant character of this representation. If it is trivial, the induced holonomy on the canonical line is trivial. Parallel transport of a nonzero canonical vector gives a global parallel volume form, which is holomorphic.

If the determinant character is nontrivial, connectedness implies that its differential is nonzero. This differential is the complex trace on the Lie algebra \mathfrak h. A compact Lie algebra is the sum of its center and its derived algebra, and trace vanishes on the latter. Hence there is a central Z\in\mathfrak h with nonzero trace. By complex irreducibility and Schur’s lemma it acts on the tangent space as a scalar ia\,\mathrm{Id}, with a\in\mathbb R nonzero. Its action on the m-fold covariant tensor power is the nonzero scalar -mia. That tensor power has no H-fixed vector. The parallelism proved in the first paragraph now excludes every nonzero holomorphic tensor of positive degree.□