Cusps and $\text{D}$-Modules

We study interactions between the categories of $\text{D}$-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 ${\text{D}}_Y$-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 ${\text{D}}_Y$ and ${\text{D}}_X$ and a Kashiwara theorem for ${\text{D}}_Y$, 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 $\text{D}$-modules on a smooth variety $X$ by collecting induced ${\text{D}}_X$-modules on varying cuspidal quotients. The resulting {\em cusp-induced} ${\text{D}}_X$-modules possess both the good properties of induced $\text{D}$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free ${\text{D}}_X$-modules.