Archimedean zeta functions, singularities, and Hodge theory

TL;DR

This paper establishes deep connections between poles of the Archimedean zeta function and singularity invariants using Hodge theory, generalizing previous results and answering longstanding questions in singularity and Hodge theory.

Contribution

It relates poles of the zeta function to singularity invariants, generalizes prior results, and provides new insights into the Hodge filtration and multiplier ideals.

Findings

Largest nontrivial pole equals negative minimal exponent of f

Counterexample showing roots of b_f(s) may not be poles of Z_f

Positive answer to minimal exponent question of Budur--Walther

Abstract

We use Hodge theory to relate poles of the Archimedean zeta function $Z_f$ of a holomorphic function $f$ with several invariants of singularities. First, we prove that the largest nontrivial pole of $Z_f$ is the negative of the minimal exponent of $f$, whose order is determined by the multiplicity of the corresponding root of the Bernstein--Sato polynomial $b_f(s)$, resolving in a strong sense a question of Musta\c{t}\u{a}--Popa. This simultaneously generalizes a result of Loeser for isolated singularities and of Koll\'ar--Litchin for the log canonical threshold, and improves them by accounting for the multiplicity. On the other hand, we give an example of $f$ where a root of $b_f(s)$ is not a pole of $Z_f$, answering a question of Loeser from 1985 in the negative. As a byproduct, we give a positive answer to a question of Budur--Walther in the case of the minimal exponent. In general, we determine poles of $Z_f$ from the Hodge filtration on vanishing cycles, sharpening a result of Barlet. Finally, we obtain analytic descriptions of the $V$-filtration of Kashiwara and Malgrange, Hodge and higher multiplier ideals, addressing another question of Musta\c{t}\u{a}--Popa. The proofs mainly rely on a positivity property of the polarization on the lowest piece of the Hodge filtration on a complex Hodge module in the sense of Sabbah--Schnell.