Stringy zeta functions for Q-Gorenstein varieties

TL;DR

This paper extends the concept of stringy invariants, like the stringy Euler number and zeta functions, to a broader class of singularities beyond log terminal, under the assumptions of the Minimal Model Program.

Contribution

It introduces new stringy invariants for almost all singularities, generalizing Batyrev's invariants to cases beyond log terminal singularities.

Findings

Stringy zeta functions are rational functions in one variable.

In the log terminal case, invariants reduce to Batyrev's stringy Euler number.

New invariants apply to singularities not strictly log canonical.

Abstract

The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities, introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties, and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the Minimal Model Program, we introduce very general stringy invariants, associated to 'almost all' singularities, more precisely to all singularities which are not strictly log canonical. They specialize to the invariants of Batyrev when the singularity is log terminal. For example the simplest form of our stringy zeta function is in general a rational function in one variable, but it is just a constant (Batyrev's stringy Euler number) in the log terminal case.