Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries

TL;DR

This paper establishes a connection between the Hilbert polynomial of Calabi-Yau hypersurfaces in toric varieties and lattice point counts in the boundary of associated polytopes, revealing a combinatorial-geometric relationship.

Contribution

It provides a novel formula expressing the Hilbert polynomial of Calabi-Yau hypersurfaces as lattice point counts in polytope boundaries, extending known results for toric varieties.

Findings

Hilbert polynomial equals lattice points in polytope boundary

Euler class computation relates to inclusion-exclusion principle

Lattice point counts in facets relate to Hilbert polynomials

Abstract

We show that the Hilbert polynomial of a Calabi-Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a convex polytope $\Delta$ is given by a lattice point count in the polytope boundary $\partial \Delta,$ just as the Hilbert polynomial of $M$ is known to be given by a lattice point count in the convex polytope $\Delta.$ Our main tool is a computation of the Euler class in $K$-theory of the normal line bundle to the hypersurface $Z,$ in terms of the Euler classes of the divisors corresponding to the facets of the moment polytope. We observe a remarkable parallel between our expression for the Euler class and the inclusion-exclusion principle in combinatorics. To obtain our result we combine these facts with the known relation between lattice point counts in the facets of $\Delta$ and the Hilbert polynomials of the smooth toric varieties corresponding to these facets.