Stringy $E$-functions of canonical toric Fano threefolds and their applications

TL;DR

This paper derives a simple combinatorial formula for the stringy E-function of 3D canonical toric Fano varieties associated with lattice polytopes with one interior point, and generalizes a known combinatorial identity.

Contribution

It provides a new combinatorial formula for the stringy E-function of certain toric Fano threefolds and extends a classical identity to a broader class of polytopes.

Findings

Derived a combinatorial formula for the stringy E-function.

Generalized the identity involving volumes of faces and dual faces.

Connected the formula to the stringy Libgober-Wood identity.

Abstract

Let $\Delta$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_{\Delta}$ associated with the polytope $\Delta$. Using the stringy Libgober-Wood identity and our formula, we generalize the well-known combinatorial identity $\sum_{\theta \preceq \Delta \atop \dim (\theta) =1} v(\theta) \cdot v(\theta^*) = 24$ holding in the case of $3$-dimensional reflexive polytopes $\Delta$.