Generalized nearby cycles via relative and logarithmic $\mathcal{D}$-modules

Abstract

For a regular map $F$ from a complex smooth affine variety $X$ to ${\mathbb{A}}_{\mathbb{C}}^r$, we construct generalized nearby-cycle modules of a regular holonomic $\mathcal{D}$-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.

83 pages. Appendix I: Relative characteristic varieties of relative regular holonomic D-modules, by Claude Sabbah