Derived Methods in Supergeometry: Fundamental Classes and Cohomology

Abstract

We apply methods from derived algebraic geometry to supergeometry. In particular, we will define super fundamental classes, which allow us to reinterpret the $\Theta_{g,n}$-classes on the moduli of curves as the pushforward of the fundamental class of the moduli of SUSY-curves. Furthermore, we study the stacky approach to cohomology in the supergeometric setting. We introduce the classic transmutation stacks (Betti, de Rham and Dolbeault) due to Simpson, which are geometric avatars of locally constant sheaves, D-modules and Higgs bundles respectively. Moreover, we refine the notion of Betti stack to allow for $P$-constructible sheaves, and we give new proofs for results due to Penkov on D-Modules and the isomorphism between de Rham cohomology and super de Rham cohomology, which one observes to be the same theorem. To do this, we will develop the theory of derived categories on superstacks establishing, amongst others, base change and recollement theorems. This is achieved via a reduction argument from the $\mathbb{Z}$-graded commutative setting, also known as Dirac geometry which allows us to deduce these theorems, without having to give the analogous proofs in the $\mathbb{Z}_2$-graded setting compared to the $\mathbb{Z}$-graded setting.