The singular support of sheaves is $\gamma$-coisotropic

Abstract

We prove that the singular support of an element in the derived category of sheaves is $\gamma$-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being $\gamma$-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by $C^1$ diffeomorphisms) and we give an example of an involutive set that is not $\gamma$-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm $\gamma$ and raise a number of new questions.