Unstable points for torus actions on flag varieties

TL;DR

This paper studies unstable points under torus actions on complex flag varieties, describing their structure and implications for cohomology vanishing, especially when the parabolic subgroup is maximal.

Contribution

It provides a detailed description of unstable points for torus actions on flag varieties and links this to cohomology vanishing results on quotient spaces.

Findings

Unstable points form a union of Schubert and opposite Schubert varieties.

Cohomology groups vanish in a range determined by the codimension of unstable points.

Explicit descriptions are given for maximal parabolic subgroups.

Abstract

In this paper, we will look at actions on complex flag varieties $G/P$ of the torus $\hat{T}=\bigcap\limits_{\alpha\in\Delta\setminus\Delta_P}\ker(\alpha)$, and under reasonable assumptions, we will give a description of the set $X^{us}$ of unstable points for $\hat{T}$-linearized invertible sheaves. We will investigate the case where $P$ is a maximal parabolic subgroup, and show that $X^{us}$ can be written as a disjoint union of a Schubert variety and an opposite Schubert variety, and we deduce the vanishing of cohomology groups $H^i(Y,{\cal M})$ for invertible sheaves ${\cal M}$ on the quotient variety $Y$ for $i$ in a range given by the codimension of $X^{us}$.