On a Bernstein-Sato polynomial of a meromorphic function

TL;DR

This paper extends the concept of Bernstein-Sato polynomials to meromorphic functions, exploring their properties, monodromies, and connections with multiplier ideals, advancing understanding in singularity theory.

Contribution

It introduces Bernstein-Sato polynomials for meromorphic functions, establishes their properties, and relates them to monodromy and multiplier ideals, providing new tools in singularity analysis.

Findings

Multiple b-functions have roots corresponding to eigenvalues of Milnor monodromies.

A Kashiwara-Malgrange type theorem for meromorphic functions is proved.

Jumping numbers of multiplier ideals relate to the new b-functions.

Abstract

We define Bernstein-Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara-Malgrange type theorem on their geometric monodromies, which would be useful also in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We introduce also multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.