Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties
Bradley Dirks
TL;DR
This work extends the Budur–Mustaţă–Saito link between multiplier ideals and the $V$-filtration to singular irreducible varieties by replacing multiplier ideals with multiplier modules and the structure sheaf with the intersection complex Hodge module ${ m IC}_X^H$. The authors develop a Bernstein–Sato theory for pairs $(X, rak a)$, establish a main equality between the multiplier module $ J(oldsymbol{ extomega}_X, rak a^{ u})$ and a $V$-filtration piece on ${ m IC}_X^H$ via graph embeddings, and derive Skoda-type results and an Ajit deformation formula in this singular setting. They also define and study a Bernstein–Sato polynomial $b_{(X, rak a)}(s)$ whose roots relate to multiplier-module jumping numbers, and show that, under a regular-sequence/ rational homology-manifold hypothesis, the absence of integer roots forces $V( rak a)$ to be a rational homology manifold. Overall, the work provides a D-module and mixed Hodge theoretic framework for singular ambient varieties, connecting log canonical thresholds, Du Bois properties, and rational homology manifold criteria with new multiplier-module and Bernstein–Sato invariants.
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.