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.
| Comments: | The proof of Theorem 3.4 is expanded, 26 pages |
| Subjects: | Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Complex Variables (math.CV) |
| Cite as: | arXiv:2412.07849 [math.AG] |
| (or arXiv:2412.07849v3 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2412.07849 arXiv-issued DOI via DataCite |
From: Ruijie Yang [view email]
[v1]
Tue, 10 Dec 2024 19:00:14 UTC (40 KB)
[v2]
Thu, 29 May 2025 17:58:57 UTC (44 KB)
[v3]
Tue, 26 May 2026 09:33:44 UTC (43 KB)
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