On `maximal' poles of zeta functions, roots of b-functions and monodromy Jordan blocks

TL;DR

This paper investigates the relationship between the maximal poles of local zeta functions, roots of b-functions, and monodromy Jordan blocks, establishing new links in singularity theory and algebraic geometry.

Contribution

It proves that maximal poles of local zeta functions correspond to roots of the Bernstein-Sato polynomial with multiplicity n, confirming a key case of a conjecture.

Findings

Maximal pole of zeta function implies a root of the Bernstein-Sato polynomial with multiplicity n.

Under certain conditions, the monodromy has a Jordan block of size n.

The results connect poles of zeta functions with monodromy and b-functions in singularity theory.

Abstract

The main objects of study in this paper are the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a polynomial or (germ of) holomorphic function in n variables. We are interested in poles of maximal possible order n. In all known cases (curves, non-degenerate polynomials) there is at most one pole of maximal order n which is then given by the log canonical threshold of the function at the corresponding singular point. For an isolated singular point we prove that if the log canonical threshold yields a pole of order n of the corresponding (local) zeta function, then it induces a root of the Bernstein-Sato polynomial of the given function of multiplicity n (proving one of the cases of the strongest form of a conjecture of Igusa-Denef-Loeser). For an arbitrary singular point we show under the same assumption that the monodromy eigenvalue induced by the pole has a Jordan block of size n on the (perverse) complex of nearby cycles.