Unstable loci in flag varieties and variation of quotients

TL;DR

This paper provides an explicit description of unstable loci in flag varieties under subgroup actions, offering a combinatorial formula, an algorithm for GIT-class determination, and applications to Mori dream spaces and rational contractions.

Contribution

It introduces a detailed combinatorial and algorithmic framework for analyzing GIT stability and quotients in flag varieties under subgroup actions, including conditions for geometric quotients and Mori dream spaces.

Findings

Unstable locus codimension varies systematically within the GIT cone.

Existence of GIT-classes with stable quotients and high codimension unstable loci.

All rational contractions are induced by GIT linearizations.

Abstract

We consider the action of a semisimple subgroup $\hat G$ of a semisimple complex group $G$ on the flag variety $X=G/B$, and the linearizations of this action by line bundles $\mathcal L$ on $X$. The main result is an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a combinatorial formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat G$-ample cone, and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension $q$ form a convex polyhedral cone. We also give a recursive algorithm for determining all GIT-classes in the $\hat G$-ample cone of $X$. As an application, we give conditions ensuring the existence of GIT-classes $C$ with an unstable locus of codimension at least two and which moreover yield geometric GIT quotients. Such quotients $Y_C$ reflect global information on $\hat G$-invariants. They are always Mori dream spaces, and the Mori chambers of the pseudoeffective cone $\overline{{\rm Eff}}(Y_C)$ correspond to the GIT-chambers of the $\hat G$-ample cone of $X$. Moreover, all rational contractions $f: Y_{C} --\to Y'$ to normal projective varieties $Y'$ are induced by GIT from linearizations of the action of $\hat G$ on $X$. In particular, this is shown to hold for a diagonal embedding $\hat G \hookrightarrow (\hat G)^k$, with sufficiently large $k$.