Archimedean zeta functions, singularities, and Hodge theory

TL;DR

This work builds a comprehensive bridge between Archimedean zeta functions and singularity invariants by embedding the problem in the framework of polarized complex Hodge modules. It proves that the largest nontrivial pole of the reduced zeta function corresponds to the negative minimal exponent $-\tilde{\alpha}_f$ with multiplicity given by the root multiplicity of $\tilde{b}_f(s)$, while showing that not every root of $b_f(s)$ yields a pole, via explicit counterexamples. The authors develop analytic descriptions of the $V$-filtration, and show how poles are read off from the Hodge filtration on vanishing cycles, yielding new characterizations of Hodge and higher multiplier ideals and providing a unified view of several previous results. They also establish positivity of the polarization on the lowest Hodge piece, which underpins the pole-order results, and supply both geometric (log-resolutions) and analytic (zeta-pole) perspectives, including a positive answer for minimal exponent questions in the Budur–Walther setting. Overall, the paper advances the interface between Hodge theory, $V$-filtration, and singularity invariants through Archimedean zeta functions, delivering precise pole descriptions, counterexamples, and analytic descriptions of multiplier-type objects.

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 Mustaţă--Popa. This simultaneously generalizes a result of Loeser for isolated singularities and of Kollár--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 Mustaţă--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.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4: Theorem \ref{['thm:lowest piece is positive']}
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Example 1.10