Support varieties of line bundle cohomology groups for SL3 (k)

TL;DR

This paper computes support varieties for line bundle cohomology groups of SL_3(k) over the first Frobenius kernel, revealing their structure and non-projectivity properties using recursive character formulas and weight regularity analysis.

Contribution

It provides explicit support variety calculations for all line bundle cohomology modules of SL_3(k) over G_1, including non-$p$-regular weights, using recursive character formulas.

Findings

Support varieties are computed for all line bundle cohomology modules.

Modules are shown to be non-projective outside the Steinberg block.

Generic dimensions do not vanish at roots of unity for $p$-regular weights.

Abstract

Let $G= SL_3(k)$ where $k$ is a field of characteristic $p > 0$ and let $\lambda \in X(T)$ be any weight with corresponding line bundle $\mathscr{L}(\lambda)$ on $G/B$. In this paper we compute the support varieties for all modules of the form $H^i(\lambda):= H^i(G/B, \mathscr{L}(\lambda))$ over the first Frobenius kernel $G_1$. The calculation involves certain recursive character formulas given by Donkin which can be used to compute the characters of the line bundle cohomology groups. In the case where $\lambda$ is a $p$-regular weight and $M=H^i(\lambda)\neq 0$ for some $i$, these formulas are used to show that any $p^{th}$ root of unity $\zeta$ is not a root of the generic dimension of $M$. To handle the case where $\lambda$ is not $p$-regular, we employ techniques similar to those used by Drupieski, Nakano and Parshall to show that the module $H^i(\lambda)$ is not projective over $G_1$ whenever it is nonzero and $\lambda$ lies outside of the Steinberg block.