Some homological properties of $GL(m|n)$ in arbitrary characteristic

TL;DR

This paper extends Penkov's super Borel-Bott-Weil theorem for $GL(m|n)$ to fields of arbitrary odd characteristic, and proves partial Kempf vanishing and characteristic-free Euler characteristic formulas.

Contribution

It generalizes key homological properties of $GL(m|n)$ to arbitrary odd characteristic fields, including super analogs of classical theorems.

Findings

Extension of Penkov's approach to arbitrary odd characteristic

Partial proof of Kempf's vanishing theorem in this setting

Characteristic-free Euler characteristic formula

Abstract

We show that Penkov's approach to a superanalog of Borel-Bott-Weil theorem for $G=GL(m|n)$ over a field of zero characteristic can be extended for a perfect field of arbitrary odd characteristic. We also prove some partial version of Kempf's vanishing theorem and characteristic free formula for Euler characteristic $\chi(B, \lambda^{\epsilon})$, where $B$ is a Borel subgroup of $G$.