Koszul's Splitting Theorem and the Super Atiyah Class

TL;DR

This paper provides a comprehensive explanation of two key results in complex supergeometry: Koszul's Splitting theorem and the super Atiyah class decomposition, highlighting their relationship through obstructions to supermanifold splitting.

Contribution

It offers a self-contained account of Koszul's theorem and the super Atiyah class decomposition, clarifying their connection in complex supergeometry.

Findings

Koszul's theorem relates to supermanifold splitting existence.

Super Atiyah class measures obstructions to splitting.

The results are interconnected through obstruction theory.

Abstract

In this article we present a self-contained account of two important results in complex supergeometry: (1) Koszul's Splitting theorem and (2) Donagi and Witten's decomposition of the super Atiyah class. These results are related in the same sense that global holomorphic connections on a holomorphic vector bundle are `related' to the Atiyah class of that vector bundle---the latter being the obstruction to the existence of the former. In complex supergeometry: Koszul's theorem pertains to the existence of supermanifold splittings whereas the super Atiyah class accordingly pertains to obstructions to the existence of splittings.