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

TL;DR

This paper extends the relationship between multiplier ideals and V-filtration to singular varieties using multiplier modules and Hodge modules, introduces a Bernstein-Sato polynomial for pairs of varieties and ideals, and explores their geometric implications.

Contribution

It generalizes the multiplier ideal and V-filtration relation to singular varieties, defines a Bernstein-Sato polynomial in this context, and links roots to geometric properties of subvarieties.

Findings

Relation between multiplier modules and V-filtration on singular varieties established

A Bernstein-Sato polynomial for pairs of varieties and ideals defined and analyzed

Roots of the polynomial linked to the rational homology manifold property

Abstract

We show that the relation between multiplier ideals and $V$-filtration on the structure sheaf due to Budur-Musta\c{t}\u{a}-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.