Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914133221310464 |
|---|---|
| author | Dirks, Bradley |
| author_facet | Dirks, Bradley |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties Dirks, Bradley Algebraic Geometry 14F10, 14B05, 14J17 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. |
| title | Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties |
| topic | Algebraic Geometry 14F10, 14B05, 14J17 |
| url | https://arxiv.org/abs/2510.25995 |