Cusps and D-modules

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 DY -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 DY and DX and a Kashiwara theorem for DY , 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 DX -modules on varying cuspidal quotients. The resulting cusp-induced DX -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 DX -modules.