On reductive automorphism groups of regular embeddings

TL;DR

This paper investigates the automorphism groups of regular embeddings of reductive groups acting on smooth varieties, identifying Levi subgroups and describing orbit structures, with applications to spherical varieties like toric and flag varieties.

Contribution

It characterizes the automorphism groups stabilizing divisors and computes invariants of the variety as a spherical A-variety, providing new structural insights.

Findings

Identified a Levi subgroup of the automorphism group stabilizing divisors.

Computed invariants of the variety as a spherical A-variety.

Described the open A-orbit and orbit closure relations.

Abstract

Let G be a connected reductive complex algebraic group acting on a smooth complete complex algebraic variety X. We assume that X under the action of G is a regular embedding, a condition satisfied in particular by smooth toric varieties and flag varieties. For any set D of G-stable prime divisors, we study the action on X of the connected automorphism group of X stabilizing D. We determine a Levi subgroup A of this automorphism group, and we compute relevant invariants of X as a spherical A-variety. As a byproduct, we obtain a description of the open A-orbit on X and the inclusion relation between A-orbit closures.