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arXiv · 2608.18003

Combinatorial Hodge Index Theorem for Polytopes

Jacob B. Wood
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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.

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Jacob B. Wood. 2026-08-18. Combinatorial Hodge Index Theorem for Polytopes. https://arxiv.org/abs/2608.18003

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