Igusa zeta functions and the non-archimedean SYZ fibration

TL;DR

This paper proves a 1999 conjecture about Igusa zeta functions' poles using the Minimal Model Program, connecting birational geometry, non-archimedean geometry, and mirror symmetry, with accessible explanations for algebraic geometers.

Contribution

It provides the first proof of Veys' conjecture, integrating techniques from birational geometry and non-archimedean geometry, and offers an accessible introduction to these advanced topics.

Findings

Proof of Veys' conjecture on poles of Igusa zeta functions

Application of the Minimal Model Program in this context

Connections established between algebraic geometry, non-archimedean geometry, and mirror symmetry

Abstract

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.