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

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Abstract

Let $\Delta$ be a $3$-dimensional lattice polytope containing exactly oneinterior lattice point. We give a simple combinatorial formula for computingthe stringy $E$-function of the $3$-dimensional canonical toric Fano variety$X_{\Delta}$ associated with $\Delta$. Using the stringyLibgober–Wood identity and our formula, we generalize the well-knowncombinatorial identity $\sum_{\substack{\theta \preceq \Delta\dim(\theta) =1}}v(\theta) \cdot v(\theta^*) = 24$ which holds for $3$-dimensional reflexive polytopes $\Delta$.

About the authors

Victor Vadimovich Batyrev

Mathematisches Institut, Universität Tübingen

Email: victor.batyrev@uni-tuebingen.de
Doctor of physico-mathematical sciences, Professor

Karin Schaller

Freie Universität Berlin

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