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
DOI:
https://doi.org/10.1090/S0894-0347-03-00439-9
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
Email:
benzvi@math.uchicago.edu, benzvi@math.utexas.edu
Thomas Nevins
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Email:
nevins@umich.edu
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