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

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 ${\sum }_{\genfrac{}{}{0ex}{}{\theta ⪯\Delta }{\dim (\theta )=1}}v(\theta )\cdot v({\theta }^{\ast })=24$ holding in the case of $3$-dimensional reflexive polytopes $\Delta$.

19 pages