Gorenstein Fano toric degenerations

TL;DR

This paper introduces a refined notion of toric degenerations that respects embeddings, establishing conditions under which Gorenstein Fano varieties degenerate to toric varieties, with implications for mirror symmetry and combinatorial geometry.

Contribution

It defines a natural framework for toric degenerations that respect embeddings and characterizes when Gorenstein Fano varieties degenerate to toric varieties, linking polytopes and duality.

Findings

Gorenstein Fano varieties degenerate to toric varieties only via their anticanonical embedding.

Reflexive polytopes are characterized by specific lattice point conditions.

Ehrhart-Macdonald Reciprocity and Serre Duality are shown to be equivalent for broad classes of varieties.

Abstract

We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polytope and the Euler characteristic of an associated torus invariant rational Weil divisor, allowing us to show that Ehrhart-Macdonald Reciprocity and Serre Duality are equivalent statements for a broad class of varieties. Additionally, we conjecture a necessary and sufficient condition for the Ehrhart quasi-polynomial of a rational convex polytope to be a polynomial. Finally, we show that the anticanonical line bundle on a Gorenstein Fano variety with at worst rational singularities is uniquely determined by a combinatorial condition of its Hilbert polynomial.