Graded covering of a supermanifold

TL;DR

This paper introduces a new concept of graded coverings for supermanifolds, extending the work of Donagi and Witten on obstruction classes, and establishes universal properties of these coverings within graded manifold categories.

Contribution

It defines and studies graded coverings of supermanifolds, generalizing the obstruction class construction and demonstrating their universal properties.

Findings

Introduces the notion of a $ ext{Z}$-graded covering for supermanifolds.

Shows that an infinite prolongation satisfies universal properties.

Connects the construction to the category of graded manifolds.

Abstract

We introduce and investigate the notion of a $\mathbb Z$-graded covering for a supermanifold. More precisely, Donagi and Witten suggested a construction of the first obstruction class for splitting of a supermanifold via differential operators. We prove that an infinite prolongation of this construction satisfies some universal properties and can be seen as a covering of a supermanifold in the category of graded manifolds.