Splitting of supervector bundles on projective superspaces

TL;DR

This paper establishes a criterion for when supervector bundles over projective superspaces split into simpler components, showing they do so under certain cohomological conditions for dimensions n ≥ 2.

Contribution

It provides a splitting criterion for supervector bundles on projective superspaces, extending understanding of their structure and identifying cases where splitting fails.

Findings

Supervector bundles with vanishing intermediate cohomology split into line bundles for n ≥ 2.

Counterexample showing non-splitting for n=1.

Criterion applies specifically to supervector bundles on projective superspaces.

Abstract

We provide a splitting criterion for supervector bundles over the projective superspaces $\mathbb{P}^{n|m}$. More precisely, we prove that a rank $p|q$ supervector bundle on $\mathbb{P}^{n|m}$ with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that $n \geq 2$. For $n=1$ we provide an example of a supervector bundle that cannot be written as a sum of line bundles.