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

Jonathan Weitsman

TL;DR

The paper proves that the Hilbert polynomial of a Calabi–Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a Delzant polytope $\Delta$ equals the lattice-point count on the boundary $\partial\Delta$, i.e., $\text{ind }\bar{\partial}_{(L^k|_Z)}=\#(k\partial\Delta\cap\mathbb{Z}^m)$. The key technique is a K-theory computation of the Euler class of the normal bundle to $Z$, expressed via the Euler classes of the facet divisors, which mirrors the inclusion-exclusion principle. By decomposing $Z$ into the intersections $D_I$ of facet divisors (each a smooth toric variety) and applying the toric Hilbert-polynomial formula to these pieces, the authors obtain the boundary-lattice count from the ambient lattice-point framework. This yields a geometric proof of the corresponding boundary-lattice-point formula and links geometric quantization to combinatorial lattice-point theory in toric Calabi–Yau settings.

Abstract

We show that the Hilbert polynomial of a Calabi-Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a convex polytope $Δ$ is given by a lattice point count in the polytope boundary $\partial Δ,$ just as the Hilbert polynomial of $M$ is known to be given by a lattice point count in the convex polytope $Δ.$ 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 $Δ$ and the Hilbert polynomials of the smooth toric varieties corresponding to these facets.

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

TL;DR

The paper proves that the Hilbert polynomial of a Calabi–Yau hypersurface in a smooth toric variety associated to a Delzant polytope equals the lattice-point count on the boundary , i.e., . The key technique is a K-theory computation of the Euler class of the normal bundle to , expressed via the Euler classes of the facet divisors, which mirrors the inclusion-exclusion principle. By decomposing into the intersections of facet divisors (each a smooth toric variety) and applying the toric Hilbert-polynomial formula to these pieces, the authors obtain the boundary-lattice count from the ambient lattice-point framework. This yields a geometric proof of the corresponding boundary-lattice-point formula and links geometric quantization to combinatorial lattice-point theory in toric Calabi–Yau settings.

Abstract

We show that the Hilbert polynomial of a Calabi-Yau hypersurface in a smooth toric variety associated to a convex polytope is given by a lattice point count in the polytope boundary just as the Hilbert polynomial of is known to be given by a lattice point count in the convex polytope Our main tool is a computation of the Euler class in -theory of the normal line bundle to the hypersurface 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 and the Hilbert polynomials of the smooth toric varieties corresponding to these facets.
Paper Structure (9 sections, 9 theorems, 62 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lattice points and toric varieties
  3. Calabi-Yau hypersurfaces in toric varieties
  4. The Euler class in $K$ theory and its uses
  5. Structure of the paper
  6. Lattice points in polytopes and Hilbert polynomials of toric varieties
  7. The Euler class in $K-$theory
  8. Hilbert polynomials of Calabi Yau hypersurfaces and lattice points in polytope boundaries: Proof of Theorem \ref{['mainthm']}
  9. Formulas for lattice points in polytope boundaries and the Proof of Theorem \ref{['bdp']}

Key Result

Theorem 1.1

The Hilbert polynomial of $M$ is given by

Figures (1)

  • Figure 1: The singular Calabi Yau $Z_{\rm sing} \subset {\mathbb CP}^2$ as a degeneration of a torus

Theorems & Definitions (12)

  • Theorem 1.1: See e.g. FultonGuillemin
  • Theorem 1.2: Khovanskii kkkkp
  • Theorem 1
  • Theorem 2: See qcy
  • Corollary 1.5
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 2 more