Geometric Invariant Theory for principal three-dimensional subgroups acting on flag varieties

TL;DR

This paper investigates Geometric Invariant Theory on flag varieties under a principal three-dimensional subgroup, explicitly classifies GIT-equivalence classes, and identifies key cones in the associated Mori dream spaces.

Contribution

It explicitly determines GIT-equivalence classes for S-ample line bundles and shows the resulting quotients are Mori dream spaces with identifiable cones.

Findings

Classification of GIT-classes for S-ample line bundles

Identification of Mori dream space structure of Hilbert quotients

Determination of pseudo-effective, movable, and nef cones

Abstract

Let G be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety G/B with respect to the action of a principal 3-dimensional simple subgroup S of G. We determine explicitly the GIT-equivalence classes of S-ample line bundles on G/B. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The Hilbert quotients Y=X//S with respect to various chambers form a family of Mori dream spaces, canonically associated with G. We are able to determine three important cones in the Picard group of any of these quotients: the pseudo-effective, the movable and the nef cones.