Generalized nearby cycles via relative and logarithmic $\mathscr{D}$-modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911725772603392 |
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| author | Wu, Lei Sabbah, with an appendix by Claude |
| author_facet | Wu, Lei Sabbah, with an appendix by Claude |
| contents | For a regular map $F$ from a complex smooth affine variety $X$ to $\mathbb A^r_\mathbb C$, we construct generalized nearby-cycle modules of a regular holonomic $\mathscr D$-module $\mathcal M$ along log strata with the log structure induced by the graph of $F$, whose relative supports are infinite unions of translated linear subvarieties of $\mathbb C^r$ determined by the zero loci of Bernstein-Sato ideals along monoid ideals. For a fixed log stratum, the nearby-cycle module corresponds to the Sabbah specialization complex of DR$(\mathcal M)$ under the relative regular Riemann-Hilbert correspondence of Fiorot-Fernandes-Sabbah, which generalizes the classical comparison theorem of Kashiwara-Malgrange for Deligne's nearby cycles. As an application, when $\mathcal M=\mathcal O_X$, we give a topological interpretation of the zero loci of Bernstein-Sato ideals of $F$ along monoid ideals under the exponential map, which answers a question of Budur-Shi-Zuo. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized nearby cycles via relative and logarithmic $\mathscr{D}$-modules Wu, Lei Sabbah, with an appendix by Claude Algebraic Geometry 14F10, 13N10, 32C38, 32S60, 32S40, 14A21 For a regular map $F$ from a complex smooth affine variety $X$ to $\mathbb A^r_\mathbb C$, we construct generalized nearby-cycle modules of a regular holonomic $\mathscr D$-module $\mathcal M$ along log strata with the log structure induced by the graph of $F$, whose relative supports are infinite unions of translated linear subvarieties of $\mathbb C^r$ determined by the zero loci of Bernstein-Sato ideals along monoid ideals. For a fixed log stratum, the nearby-cycle module corresponds to the Sabbah specialization complex of DR$(\mathcal M)$ under the relative regular Riemann-Hilbert correspondence of Fiorot-Fernandes-Sabbah, which generalizes the classical comparison theorem of Kashiwara-Malgrange for Deligne's nearby cycles. As an application, when $\mathcal M=\mathcal O_X$, we give a topological interpretation of the zero loci of Bernstein-Sato ideals of $F$ along monoid ideals under the exponential map, which answers a question of Budur-Shi-Zuo. |
| title | Generalized nearby cycles via relative and logarithmic $\mathscr{D}$-modules |
| topic | Algebraic Geometry 14F10, 13N10, 32C38, 32S60, 32S40, 14A21 |
| url | https://arxiv.org/abs/2602.05314 |