Unstable points for torus actions on flag varieties

Abstract

In this paper, we will look at actions on complex flag varieties $G/P$ of the torus $\hat{T}=\bigcap _{\alpha \in \Delta \setminus {\Delta }_P}\ker (\alpha )$, 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 $H^i(Y,\mathcal{M})$ for invertible sheaves $\mathcal{M}$ on the quotient variety $Y$ for $i$ in a range given by the codimension of $X^{us}$.

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