On the Geometry of Forms on Supermanifolds

TL;DR

This paper rigorously explores the geometry of differential and integral forms on supermanifolds, establishing cohomology, sheaf properties, and integral theorems, with new proofs and perspectives on supergeometry.

Contribution

It introduces new algebraic-geometric approaches to forms on supermanifolds, including cohomology computations, sheaf characterizations, and a novel perspective on pseudoforms.

Findings

Cohomology of the de Rham complex computed

Superanalog of Stokes' theorem proved

Integral forms are shown to be quasi-isomorphic to differential forms

Abstract

This paper provides a rigorous account on the geometry of forms on supermanifolds, with a focus on its algebraic-geometric aspects. First, we introduce the de Rham complex of differential forms and we compute its cohomology. We then discuss three intrinsic definitions of the Berezinian sheaf of a supermanifold - as a quotient sheaf, via cohomology of the super Koszul complex or via cohomology of the total de Rham complex. Further, we study the properties of the Berezinian sheaf, showing in particular that it defines a right $\mathcal{D}$-module. Then we introduce integral forms and their complex and we compute their cohomology, by providing a suitable Poincar\'e lemma. We show that the complex of differential forms and integral forms are quasi-isomorphic and their cohomology computes the de Rham cohomology of the reduced space of the supermanifold. The notion of Berezin integral is then introduced and put to the good use to prove the superanalog of Stokes' theorem and Poincar\'e duality, which relates differential and integral forms on supermanifolds. Finally, a different point of view is discussed by introducing the total tangent supermanifold and (integrable) pseudoforms in a new way. In this context, it is shown that a particular class of integrable pseudoforms having a distributional dependence supported at a point on the fibers are isomorphic to integral forms. Within the general overview several new proofs of results are scattered.