Combinatorial Hodge Index Theorem for Polytopes

Abstract: Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology of the associated toric variety when the polytope is rational. Maxim-Schuermann computed the intersection cohomology signature of a projective toric variety, corresponding to the case of a polytope with rational vertices. Using the combinatorial framework of BBFK, we show that the Maxim-Schuermann formula extends to arbitrary convex polytopes. Finally, we discuss a version of the Hodge index theorem for polytopes.

• The paper proves that the intersection-pairing signature of a projective polytope equals a face-wise sum of Stanley g-polynomial evaluations, extending the Maxim–Schürmann formula to non-rational polytopes.
• The proof uses combinatorial intersection cohomology, hard Lefschetz, and Hodge–Riemann relations to derive the signature from intersection-cohomology Betti numbers without invoking toric varieties or mixed Hodge modules.
• The result identifies the signature with a combinatorial intersection-cohomology χ₁-genus, while leaving open whether analogous Lefschetz operators and signature formulas exist for non-projective fans.

Overview and main result

This paper, by Jacob B. Wood, establishes a purely combinatorial formula for the signature of the intersection cohomology pairing of a convex polytope, thereby extending the Maxim–Schürmann signature formula for projective toric varieties to arbitrary (possibly non-rational) convex polytopes (2608.18003). The central result is that for a projective fan $\Phi$ of dimension $n$ associated to a convex polytope,

$$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$

where $g_\Delta(t)$ is Stanley's $g$-polynomial of the proper face $\Delta$. The proof is entirely combinatorial: it uses the hard Lefschetz theorem and Hodge–Riemann relations for combinatorial intersection cohomology, avoiding mixed Hodge modules, characteristic classes, and any reference to an associated toric variety. This is significant because there exist polytopes with no combinatorially equivalent realization having rational coordinates [Z], so no toric variety is available in general.

Background: combinatorial intersection cohomology

The paper works within the framework introduced by Barthel–Brasselet–Fieseler–Kaup (BBFK). For a complete fan $\Phi$ in a real vector space $V$, one considers the sheaf $\mathcal{A}_\Phi$ of piecewise polynomial functions on the poset of cones. The minimal extension sheaf $\mathcal{L}_\Phi$—unique up to isomorphism, characterized by normalization along cone boundaries—has global sections whose quotient by the maximal ideal $n$0 defines the combinatorial intersection cohomology:

$n$1

When $n$2 is the normal fan of a rational polytope, this recovers the equivariant and non-equivariant intersection cohomology of the associated projective toric variety. BBFK showed this module carries a non-degenerate Poincaré pairing even for non-rational polytopes; for simplicial fans, Brion gave the explicit description $n$3.

The Kähler package and the signature computation

A strictly convex piecewise linear function on $n$4—which exists precisely when $n$5 is the normal fan of a convex polytope, i.e., when $n$6 is projective—acts as a Lefschetz operator. Karu proved the hard Lefschetz theorem ($n$7 is an isomorphism), which forces symmetry and unimodality of the Betti numbers $n$8; Bressler–Lunts established the Hodge–Riemann relations, stating that the quadratic form $n$9 is positive definite on each primitive subspace $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$0.

The signature computation follows the classical Hodge index argument. When $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$1 is odd, all odd-degree intersection cohomology vanishes, so $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$2 and $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$3. When $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$4 is even, hard Lefschetz gives the orthogonal primitive decomposition

$$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$5

and Hodge–Riemann determines the sign of each summand, yielding

$$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$6

Reindexing and using Lefschetz symmetry collapses this to the alternating sum $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$7. The final step identifies these Betti numbers with Stanley's combinatorial invariants: Bressler–Lunts' recursive relations among local $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$8- and $$\sigma(\Phi)=\sum_{\Delta\prec\Phi} g_\Delta(-1)(-2)^{n-\dim\Delta-1},$$9-polynomials coincide exactly with Stanley's recursions for $g_\Delta(t)$0- and $g_\Delta(t)$1-polynomials under the identification of the poset of cones with the face lattice, giving $g_\Delta(t)$2 and $g_\Delta(t)$3. Substituting produces the stated formula. Notably, the paper notes a sign convention subtlety: relative to Maxim–Schürmann's notation, the result applies to their polar polytope, accounting for differing powers of $g_\Delta(t)$4.

A combinatorial Hodge index theorem

Since intersection cohomology of a projective toric variety is of Hodge–Tate type (concentrated in bidegrees $g_\Delta(t)$5), the paper defines a formal bigrading on combinatorial intersection cohomology by declaring $g_\Delta(t)$6 if $g_\Delta(t)$7 and zero otherwise. With this convention, the intersection cohomology Hodge $g_\Delta(t)$8-polynomial satisfies

$g_\Delta(t)$9

i.e., it coincides with the toric $g$0-polynomial evaluated at $g$1. Specializing at $g$2 yields the combinatorial analog of the Hodge index theorem:

$g$3

In the geometric setting this is consistent with the identity $g$4 for possibly singular projective toric varieties, known more generally for arbitrary projective varieties (Maxim–Saito–Schürmann) and compact varieties (Fernández de Bobadilla–Pallarès–Saito).

Limitations and open questions

The paper is candid about scope restrictions. First, the signature formula requires $g$5 to be projective, since the Lefschetz operator comes from a strictly convex piecewise linear function; the author states explicitly that it is unknown whether such an operator exists for general fans, leaving the non-projective case open. Second, the "Hodge structure" imposed on combinatorial intersection cohomology is definitional rather than derived from geometry—it is an ansatz mimicking the Hodge–Tate property—and its naturality beyond the projective setting is not addressed. Third, the extension to arbitrary projective varieties or to non-rational polytopes admitting other geometric interpretations lies outside the paper's scope.

Conclusion

The paper gives a self-contained combinatorial proof that the signature of the intersection pairing of a projective fan equals an alternating evaluation of Stanley's $g$6-polynomials over faces, recovering the Maxim–Schürmann formula without geometric input and extending it to non-rational polytopes. By packaging the result as $g$7 via a formal Hodge–Tate bigrading, it provides a clean combinatorial counterpart to the classical Hodge index theorem. The main open question left by the work is whether a Lefschetz operator—and hence a Kähler package and signature theory—exists for non-projective fans.