dg-Methods for Microlocalization

Abstract

For a complex manifold $X$ the ring of microdifferential operators ${\mathcal{E}}_X$ acts on the microlocalization $\mu hom(F,{\mathcal{O}}_X)$ for $F$ in the derived category of sheaves on $X$. Kashiwara, Schapira, Ivorra, Waschkies proved as a byproduct of their new microlocalization functor for ind-sheaves, ${\mu }_X$, that $\mu hom(F,{\mathcal{O}}_X)$ can in fact be defined as an object of $\mathrm{D}({\mathcal{E}}_X)$: this follows from the fact that ${\mu }_X{\mathcal{O}}_X$ is concentrated in one degree.
In this paper we prove that the tempered microlocalization $T-\mu hom(F,{\mathcal{O}}_X)$ and in fact ${\mu }_X{\mathcal{O}}_X^t$ also are objects of $\mathrm{D}({\mathcal{E}}_X)$. Since we don't know whether ${\mu }_X{\mathcal{O}}_X^t$ is concentrated in one degree we built resolutions of ${\mathcal{E}}_X$ and ${\mu }_X{\mathcal{O}}_X^t$ such that the action of ${\mathcal{E}}_X$ is realized in the category of complexes (and not only up to homotopy). To define these resolutions we introduce a version of the de Rham algebra on the subanalytic site which is quasi-injective. We prove that some standard operations in the derived category of sheaves can be lifted to the (non-derived) category of dg-modules over this de Rham algebra. Then we built the microlocalization in this framework.