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.