Stringy Hodge numbers of varieties with Gorenstein canonical singularities

TL;DR

This paper introduces stringy E-functions for varieties with mild singularities, computes them for toric varieties, and proposes a method to define stringy Hodge numbers, enabling mirror symmetry tests for singular Calabi-Yau varieties.

Contribution

It develops a new framework for defining stringy Hodge numbers for varieties with Gorenstein canonical singularities, extending mirror symmetry tools to singular cases.

Findings

Explicit computation of stringy E-functions for toric varieties

Proposal of a general method for stringy Hodge numbers

Application to topological mirror symmetry tests

Abstract

We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers.