The invariant index and its value at zero

Let \(T\) be a compact real torus acting smoothly by holomorphic automorphisms on a compact complex manifold \(F\). Its differential acts on the anticanonical line \(L_F=-K_F=\det T_F\). We call this the natural linearization. The corresponding action on a section is \((t\cdot s)(x)=t_{L_F}s(t^{-1}x)\); tensor powers carry the induced action. For each integer \(m\ge0\), define \[I_T(m)=\sum_{q\ge0}(-1)^q\dim H^q(F,L_F^m)^T.\] Thus \(I_T(m)\) is the multiplicity of the trivial representation in the Dolbeault index of \(L_F^m\).

The theorem has no projectivity, Kählerness, or positivity premise. Its endpoint is essential: when \(I_T(0)\ne0\), the polynomial cannot vanish at all large divisible exponents. Some fixed cohomological degree therefore contains invariant classes at unbounded exponents. Eventual polynomiality alone would give no information about whether that polynomial is zero.

The natural linearization is equally essential. On a point, twisting the trivial line by a nonzero torus character gives invariant dimension zero at every positive power and dimension one at power zero. For the natural anticanonical action, by contrast, the fiber character at a fixed component is the sum of the normal tangent characters. This identity controls the endpoint.

Our calculation uses the holomorphic Lefschetz framework of Atiyah–Bott (Atiyah and Bott 1966) and its compact-group, fixed-component form due to Atiyah–Segal and Atiyah–Singer (Atiyah and Segal 1968; Atiyah and Singer 1968). Polarizing its denominators expresses the invariant multiplicity as polynomially weighted lattice counts on rational polytopes with specified coordinate faces removed. Weighted Ehrhart theory supplies polynomiality on divisible exponents (Baldoni et al. 2012, preprint §2.4 and §3.1). To determine the value at zero, we prove the count directly, retaining all small divisible exponents, and then show that the anticanonical character gives a deformation retraction onto the removed faces. The cancellation is a difference of Euler characteristics of compact polyhedra. Stanley’s treatment of deleted boundary already identifies this constant term and singles out boundary visible from an exterior point (Stanley 1974, Proposition 8.2 and the discussion and proof of Proposition 8.3). Here the anticanonical character produces such an exterior point in the affine counting space. We give a direct weighted argument and an explicit retraction, including degenerate polytopes.

When \(F\) is Kähler and \(L_F\) has a smooth semipositive metric, hard Lefschetz with semipositive coefficients (Demailly et al. 2001, Theorem 0.1) gives \[H^0(F,\Omega_F^{n-q}\otimes L_F^{m+1}) \longrightarrow H^q(F,L_F^m),\qquad n=\dim F,\] surjectively. An invariant Kähler form makes this map equivariant; compact averaging then preserves surjectivity on invariant subspaces. The coefficient is \(L_F^{m+1}\) because the cohomological target is \(K_F\otimes L_F^{m+1}=L_F^m\). Thus Theorem 1 supplies invariant twisted differential forms when \(I_T(0)\ne0\). The use of hard Lefschetz and a nonzero Euler characteristic to obtain infinitely many twisted forms already appears for canonical powers in (Demailly et al. 2001, Theorem 2.7.3); here the invariant anticanonical index requires a separate argument to retain its value at exponent zero.

Removing a fixed error

We use additive notation for line bundles and Cartier divisors. A divisor is pseudoeffective if its numerical class lies in the closure of the effective cone. Smooth semipositivity means the existence of a smooth Hermitian metric with nonnegative Chern curvature; it implies nefness. The next theorem is an ordinary section theorem, with no group action.

The fixed error need not be effective. No rational-connectedness or Euler-characteristic assumption on \(S\) is used. The argument begins with a rational map whose base has maximal dimension among maps \(S\dashrightarrow Y\) for which a multiple of \(L\) dominates an ample base divisor in pseudoeffective order. Maximality forces ratios of the relevant sections to be base functions. Their horizontal zero orders consequently vary affinely with the exponent.

After birational modifications, a normalization along base divisors produces a line bundle \(B\) on a smooth base \(Y\) and a nonzero map \[\sigma:f^*B\longrightarrow\ell\pi^*L,\qquad \ell>0,\] where \(V\) is smooth, \(\pi:V\to S\) is birational, and \(f:V\to Y\) is a morphism. This map will be used to transfer the sections constructed on the base to positive multiples of \(L\) on \(S\). The same maximality gives rank-one adjoint direct images, whose integral metrics yield a semipositive metric on \(B\) by Berndtsson’s positivity theorem (Berndtsson 2009). The normalization of \(\sigma\) makes the resulting weights locally bounded, including near singular fibers.

These bounds allow Fujino’s Kollár–Nadel theorem (Fujino 2018) to give vanishing at every nonnegative base twist. A single Euler polynomial therefore equals the corresponding section dimension even at zero, where the exceptional canonical section makes it positive. Some positive exponent gives a section, which \(\sigma\) transfers back to \(S\). As in the invariant-index argument, control at zero is what rules out the zero polynomial; the two constructions of that control are independent.

This criterion applies to differential forms through a determinant construction. The generic-span method of Lazić–Peternell (Lazić and Peternell 2018, Lemma 4.1), adapted in Lazić–Matsumura–Peternell–Tsakanikas–Xie (Lazić et al. 2023, Lemma 5.1), extracts one fixed determinant line from infinitely many twisted forms. We give the argument for an unbounded positive sequence, including the numerically trivial case excluded by the hypotheses of those two cited lemmas. The cotangent subsheaf theorem (Lazić et al. 2023, Theorem 4.1), with Ou’s generic nefness result (Ou 2023, Theorem 1.4) as an antecedent, makes the negative of that line pseudoeffective. Theorem 2 then removes the error.

Application through monodromy

For a smooth connected projective \(X\), anticanonical nonvanishing asks for a section of \(-mK_X\) for some \(m>0\). The positive-multiple question under smooth semipositivity is associated with Yau’s anticanonical-section problem (Yau 1994, Problem 75). The prescribed-Ricci theorem (Yau 1978) and the structure theorems of Demailly–Peternell–Schneider and Campana–Demailly–Peternell (Demailly et al. 1996; Campana et al. 2015) identify a compact rationally connected factor of the universal cover. Residual monodromy can still act on it through a compact torus, so a section on that factor alone need not descend. Müller’s equivariant theorem produces invariant plurisections under semiampleness (Müller 2025, Theorem C). His Theorem A proves nonvanishing for projective klt pairs with nef anti-log-canonical divisor when its restriction to the general fiber of the maximal rationally connected fibration is semiample; Corollary B proves nonvanishing for projective klt threefold pairs with nef anti-log-canonical divisor. Our smooth application assumes metric semipositivity and imposes no fiber-semiampleness condition.

We apply the invariant index on the compact factor \(F\) and ordinary conversion on a finite cover \(S\) of \(X\); descending forms is the step that connects those two spaces.

The companion Anticanonical nonvanishing from smooth semipositivity (OpenAI 2026) proves a finite-cover description with the residual torus and invariant canonical frames retained, as well as the finite étale norm of a section. We use only these geometric inputs. We prove locally that invariant twisted forms on the compact factor descend injectively to the finite cover. Theorem 1 and hard Lefschetz provide those forms, and Theorem 2 is applied after descent. Taking the norm then proves that \(-mK_X\) has a nonzero section for some \(m>0\).

Passing to a positive multiple is necessary. On an Enriques surface \(E\), \(K_E\) has order two and \(H^0(E,-K_E)=0\) (Dolgachev 2016). Consequently \(E\times\mathbb P^1\) has a smoothly semipositive, non-torsion anticanonical bundle without a first-power section. In dimension two, the classification of Chen–Filip–Sun–Tosatti–Zhang (Chen et al. 2026, Theorem 1.3) gives complementary geometric context.¹

Section 2 proves the index theorem and derives invariant twisted forms. Section 3 proves conversion by constructing the normalized relative section and bounded base metric. Section 4 gives the determinant argument. Section 5 proves form descent and the global application. The two principal theorems are independent of one another; their combination takes place only in that last application.

The deleted boundary

Polarizing a fixed-point denominator may exclude some coordinate faces. The next lemma proves the cancellation needed for those exclusions. Its geometry is simple: a point outside the nonnegative orthant lies in the affine space containing the polytope, and rays from that point first enter the polytope through the faces being removed.

Figure 1 illustrates this retraction when all three vectors \(\alpha_i\) are \(1\) and \(I=\{1\}\). The proof above also applies when \(D\) has smaller dimension or consists of one point.

Localization and the anticanonical character

We now identify the lattice domains in the fixed-point formula. The two lemmas above will determine their constant terms component by component.

Invariant twisted forms

The index theorem is independent of positivity. For its geometric application we now assume Kählerness and introduce the analytic map that turns cohomology into holomorphic forms.

This is the smooth-metric case of (Demailly et al. 2001, Theorem 0.1); its multiplier ideal is trivial. The smooth-coefficient Lefschetz theorem of Mourougane (Mourougane 1999, Theorem 2.6) and the earlier nef-coefficient cohomology work of Takegoshi (Takegoshi 1997, Theorem 1) are antecedents of this statement. We use the formulation in (Demailly et al. 2001). If a compact group preserves the data, the map is equivariant. Averaging over the group then preserves surjectivity on invariant subspaces.

The base and the relative section

The bounded base metric

We have constructed \(B\) and \(\sigma\); the next task is a semipositive metric on \(B\) with locally bounded weights. For a local frame \(\beta\), our weight convention is \(-\log|\beta|^2\), so semipositive curvature means that the weights are plurisubharmonic. We obtain them as limits of adjoint integral metrics. We use Berndtsson’s direct-image theorem in the following form (Berndtsson 2009, Theorem 1.2): for a proper holomorphic submersion with Kähler total space and a smoothly semipositive line bundle \(A\), the natural \(L^2\) metric on \(f_*(K_{V/Y}+A)\) has semipositive curvature wherever the adjoint sections form a vector bundle. The coefficient metric may be semipositive; strict positivity along the fibers is not required.

Let \(h_L\) denote the pulled-back smooth metric on \(\pi^*L\). Choose a dense Zariski open \(Y^\circ\) on which \(f\) is smooth and \(e\) and \(\sigma\) are not identically zero on any fiber. The fibers there are connected: relative algebraic closedness of \(\mathbb C(Y)\) in \(\mathbb C(S)\) makes the finite part of the Stein factorization trivial. For a local frame \(\beta\) of \(B\), set \(\sigma_\beta=\sigma(f^*\beta)\) and \[ \varphi_\beta(y)=-\log\max_{V_y} |\sigma_\beta|_{h_L^\ell}^{\,2},\qquad y\in Y^\circ. \tag{12}\] The fiberwise maximum is positive and continuous on this open set.

To see that \(\varphi_\beta\) is plurisubharmonic, take a local frame \(\tau\) of \(K_Y\). Equation (10) gives \[ K_{V/Y}+(1+k\ell)\pi^*L =E+k\ell\pi^*L-f^*K_Y. \tag{13}\] Thus \(e\sigma_\beta^k/f^*\tau\) is an adjoint section. By (11) and generic base change, it spans the adjoint direct image on a dense open subset of \(Y^\circ\). For every integer \(k\ge1\), Berndtsson’s theorem makes \[ \varphi_{\beta,k}(y) =-\frac1k\log\int_{V_y} |\sigma_\beta|_{h_L^\ell}^{\,2k} \left|e/f^*\tau\right|_{h_L}^{\,2} \tag{14}\] plurisubharmonic on that dense open. The integral is smooth and strictly positive throughout \(Y^\circ\); its curvature inequality therefore holds on all of \(Y^\circ\) by continuity.

The final factor in the integral is the smooth nonnegative measure obtained from a \(\pi^*L\)-valued relative canonical form. It has positive mass and full support on each fiber, because a nonzero holomorphic section on a connected smooth fiber cannot vanish on an open subset. Denote this measure by \(\mu_y\), its mass by \(M(y)>0\), and put \[q_\beta=|\sigma_\beta|_{h_L^\ell}^{\,2},\qquad A_k(y)=\left(\frac1{M(y)}\int_{V_y}q_\beta^k\,d\mu_y\right)^{1/k}.\] Hölder’s inequality for the probability measure \(M(y)^{-1}\mu_y\) shows that \(A_k(y)\) increases with \(k\). Full support gives \(A_k(y)\to\max_{V_y}q_\beta\). The moments and their limit are continuous, so Dini’s theorem makes this convergence uniform on every compact subset of a coordinate open in \(Y^\circ\). The limit is positive there. Since \[\varphi_{\beta,k}=-\log A_k-\frac1k\log M,\] the weights in (14) converge locally uniformly to \(\varphi_\beta\). Their limit is plurisubharmonic.

It remains to establish local bounds near the omitted fibers. Properness of \(f\) and smoothness of the upstairs metric bound \(q_\beta\) above over every relatively compact base neighborhood. Equation (12) therefore gives a local lower bound for \(\varphi_\beta\), even as one approaches the boundary.

For the upper bound, let \(P\) be a divisorial component of \(Y\setminus Y^\circ\). The normalization of \(\sigma\) supplies a component \(Q\) above \(P\) on which \(\sigma\) is generically nonzero. Choose a general point of \(Q\) where \(P,Q\) are smooth, \(Q\to P\) is submersive, no other component of \(f^*P\) passes through the point, and \(\sigma\ne0\). In suitable local coordinates, a transverse coordinate to \(P\) pulls back to a unit times \(z_1^r\), with \(r>0\), while the other base coordinates pull back to independent coordinates along \(Q\). Taking a local root absorbs the unit. This local form shows that a small source neighborhood maps onto a base neighborhood. On a still smaller source neighborhood, \(q_\beta\) has a positive lower bound. Every sufficiently nearby fiber meets this neighborhood, so its maximum has the same lower bound. This gives an upper bound for \(\varphi_\beta\) near a general point of \(P\).

The removable-singularity theorem for plurisubharmonic functions now extends \(\varphi_\beta\) across dense open subsets of all boundary divisors. These opens can be taken Zariski open: nonvanishing and the required differential ranks hold on algebraic opens of \(Q\), whose dominant constructible images contain dense Zariski opens of \(P\). The set still omitted is therefore contained in a closed analytic set \(A\subset Y\) of codimension at least two.

We recall why no upper-bound obstruction remains at \(A\). Near a point of \(A\), choose a small affine complex disk centered there whose boundary misses \(A\), and a compact neighborhood of that boundary disjoint from \(A\). The plurisubharmonic function has a common upper bound on this neighborhood. For every nearby \(y\notin A\), choose a disk of the same radius centered at \(y\), with direction sufficiently close to the original one that its boundary stays in this fixed neighborhood. Its direction can also be chosen so that the whole disk misses \(A\): the radial image of \(A\), viewed from \(y\), in the space of complex line directions has real dimension at most \(2\dim Y-4\), smaller than the direction space dimension \(2\dim Y-2\). The submean inequality gives a common upper bound at all such \(y\). Removable singularities therefore extend the function across \(A\) as well. The upper-limit extension preserves the lower bound already supplied by properness.

We have obtained locally bounded plurisubharmonic weights on all of \(Y\). Under \(\beta'=g\beta\), formula (12) gives \[\varphi_{\beta'}=\varphi_\beta-\log|g|^2.\] The same relation holds after extension, so the weights define a semipositive singular Hermitian metric \(h_B\) on \(B\), with locally bounded weights.

Vanishing at every nonnegative exponent

The bounded metric on \(B\) is now available. To produce sections, we combine it with a metric on \(\pi^*L\) that is positive in base directions and has no multiplier-ideal loss. Recall that \(\mathcal J(h)\) consists locally of holomorphic functions \(g\) for which \(|g|^2e^{-\varphi_h}\) is integrable.

The domination (6) gives a singular metric on \(\pi^*L\) whose curvature is positive in base directions. More precisely, combine a semipositive singular metric on \(c\pi^*L-f^*H\) with a positive smooth metric on \(H\), and take the \(c\)-th root. The resulting metric \(h_1\) satisfies \[\Theta_{h_1}(\pi^*L)\ge c^{-1}f^*\omega_H.\] Mix its local weights with those of the smooth semipositive metric \(h_L\): for some \(0<\delta<1\), put \(h=h_L^{1-\delta}h_1^\delta\). The local exponential-integrability consequence of Skoda’s theorem (Skoda 1972, Proposition 7.1, p. 389) supplies, near each point, a sufficiently small positive exponent for which the negative exponential of the singular weight is integrable. We use its arbitrary-plurisubharmonic-weight formulation in (Demailly and Kollár 2001, Definition 0.1 and 1.4(8)): for a nontrivial psh weight \(\varphi\), the local integrability threshold of \(e^{-2\gamma\varphi}\) is positive. This formulation also includes complex dimension one. In our squared-norm convention, take \(\gamma=\delta/2\). A finite coordinate cover of the compact space \(V\) gives one common \(\delta>0\). The smooth part changes integrability by bounded factors. Hence \[ \Theta_h(\pi^*L)\ge\epsilon f^*\omega_H, \qquad \mathcal J(h)=\mathcal O_V, \qquad \epsilon=\delta/c>0. \tag{15}\]

For each integer \(k\ge0\), tensor this metric with \(f^*h_B^k\). Its curvature is still at least \(\epsilon f^*\omega_H\), and its multiplier ideal is still trivial: for every fixed \(k\), the added weight is locally bounded. Thus the same choice of \(h\) works at all the required exponents.

We use Fujino’s Kollár–Nadel vanishing theorem in the following form (Fujino 2018, Theorem 1.3). If \(f:V\to Y\) is a surjective morphism from a compact Kähler manifold to a projective variety and a line bundle \(A\) has a singular metric \(h_A\) with \(\Theta_{h_A}(A)\ge\epsilon f^*\omega_H\), then \[H^i\bigl(Y,R^qf_*(\mathcal O_V(K_V+A)\otimes\mathcal J(h_A))\bigr)=0 \qquad (i>0,\ q\ge0).\] Fujino derives this form from the injectivity and multiplier-ideal Bertini theorems of Fujino–Matsumura (Fujino 2018, Theorems 1.1–1.3). The metric need not have analytic singularities. Apply the theorem to \(A=\pi^*L+kf^*B\), with the metric just constructed, and take \(q=0\). Since \(K_V+\pi^*L=E\), the projection formula gives \[ H^i\bigl(Y,\mathcal G\otimes\mathcal O_Y(kB)\bigr)=0 \qquad(i>0,\ k\ge0), \qquad \mathcal G=f_*\mathcal O_V(E). \tag{16}\]

Riemann–Roch on the smooth projective base makes \[Q(k)=\chi\bigl(Y,\mathcal G\otimes\mathcal O_Y(kB)\bigr) =\int_Y\mathop{\mathrm{ch}}(\mathcal G)e^{k c_1(B)}\mathop{\mathrm{td}}(Y)\] a polynomial in \(k\), whether or not \(B\) is ample. By (16), this polynomial equals \(h^0(Y,\mathcal G\otimes\mathcal O_Y(kB))\) at every nonnegative integer. At zero the canonical section \(e\) gives a nonzero section, so \(Q(0)>0\). This is why vanishing at \(k=0\) is indispensable: without it, the existence of \(e\) would not determine the Euler characteristic. The polynomial is nonzero, and hence cannot vanish at every positive integer. For some \(k>0\) there is a nonzero section of \(E+kf^*B\) on \(V\).

Multiply this section by \(\sigma^k\) from (9). The result is a nonzero section of \(E+k\ell\pi^*L\). Finally, \(\pi_*\mathcal O_V(E)=\mathcal O_S\): a rational function with possible poles only on the effective exceptional divisor \(E\) is regular away from its codimension-two image, and extends across that image by normality of \(S\). The projection formula therefore sends our section to a nonzero section of \(k\ell L\), completing the proof of Theorem 2.

The two appearances of \(L=-K_S\) explain the strength and the scope of the criterion. The identity \(K_V+\pi^*L=E\) supplies the adjoint section used in the integral metric and turns adjoint vanishing into the fixed sheaf \(f_*\mathcal O_V(E)\) at exponent zero. These steps connect the positive base metric to an Euler polynomial with a known nonzero value.