Cusps and 𝒟-modules

Ben-Zvi, David; Nevins, Thomas

doi:10.1090/S0894-0347-03-00439-9

Cusps and $\mathcal {D}$-modules
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by David Ben-Zvi and Thomas Nevins

J. Amer. Math. Soc. 17 (2004), 155-179

DOI: https://doi.org/10.1090/S0894-0347-03-00439-9

Published electronically: September 24, 2003

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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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