Abstract: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.
From: Lei Wu [view email]
[v1]
Thu, 5 Feb 2026 05:20:43 UTC (57 KB)
[v2]
Sun, 1 Mar 2026 18:32:33 UTC (57 KB)
[v3]
Thu, 28 May 2026 00:50:38 UTC (67 KB)
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