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arXiv · 2510.25995

Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties

Bradley Dirks
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Abstract

We show that the relation between multiplier ideals and $V$-filtration on the structure sheaf due to Budur-Mustaţă-Saito generalizes to singular irreducible varieties, by replacing multiplier ideals with multiplier modules and the structure sheaf with the intersection complex Hodge module. This is applied to a Skoda theorem for such modules as well as a $\mathcal D$-module theoretic proof of Ajit's formula relating the multiplier modules of an ideal to those of the Rees parameter in the extended Rees algebra. Moreover, we define a Bernstein-Sato polynomial for the pair of a variety and an ideal sheaf on it. We relate the roots to the jumping numbers of the multiplier modules. If the ideal is generated by a regular sequence on a rational homology manifold, we show that the absence of integer roots of the polynomial implies that the subvariety defined by the ideal is a rational homology manifold.

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Bradley Dirks. 2025-11-04. Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties. https://arxiv.org/abs/2510.25995

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