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Cusps and $\mathcal{D}$-modules

Authors: David Ben-Zvi and Thomas Nevins
Journal: J. Amer. Math. Soc. 17 (2004), 155-179
MSC (2000): Primary 14F10, 13N10, 16S32, 32C38
Published electronically: September 24, 2003
MathSciNet review: 2015332
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Abstract: We study interactions between the categories of $\mathcal{D}$-modules on smooth and singular varieties. For a large class of singular varieties $Y$, we use an extension of the Grothendieck-Sato formula to show that $\mathcal{D}_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the cuspidal quotients. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\mathcal{D}_Y$ and $\mathcal{D}_X$and a Kashiwara theorem for $\mathcal{D}_Y$, thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing results for complex curves and surfaces and rational Cherednik algebras). We also use this equivalence to enlarge the category of induced $\mathcal{D}$-modules on a smooth variety $X$ by collecting induced $\mathcal{D}_X$-modules on varying cuspidal quotients. The resulting cusp-induced $\mathcal{D}_X$-modules possess both the good properties of induced $\mathcal{D}$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free $\mathcal{D}_X$-modules.

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Additional Information

David Ben-Zvi
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, University of Texas, Austin, Texas 78712-0257

Thomas Nevins
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Keywords: ${\mathcal D}$-modules, Grothendieck-Sato formula, Morita equivalence
Received by editor(s): December 6, 2002
Published electronically: September 24, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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