{
  "id": "math/0212094",
  "version": "v3",
  "published": "2002-12-05T21:51:27.000Z",
  "updated": "2003-10-09T18:45:36.000Z",
  "title": "Cusps and $\\D$-Modules",
  "authors": [
    "David Ben-Zvi",
    "Thomas Nevins"
  ],
  "comment": "Final version, to appear in J. Amer. Math. Soc. (2004)",
  "categories": [
    "math.AG",
    "math.AC",
    "math.RA"
  ],
  "abstract": "We study interactions between the categories of $\\D$-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 $\\D_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the {\\em cuspidal quotients}. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\\D_Y$ and $\\D_X$ and a Kashiwara theorem for $\\D_Y$, 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 $\\D$-modules on a smooth variety $X$ by collecting induced $\\D_X$-modules on varying cuspidal quotients. The resulting {\\em cusp-induced} $\\D_X$-modules possess both the good properties of induced $\\D$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free $\\D_X$-modules.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2003-10-09T18:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular varieties",
      "rational cherednik algebras",
      "grothendieck-sato formula",
      "smooth bijective normalization",
      "morita equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....12094B"
    }
  }
}