Automorphism groups of toroidal horospherical varieties

TL;DR

This paper characterizes the automorphism groups of smooth complete toroidal horospherical varieties, providing criteria for reductivity and applying results to K-unstability of certain fiber bundles.

Contribution

It offers a structure theorem for automorphism groups of these varieties and introduces a criterion for their reductivity, advancing understanding of their symmetry properties.

Findings

Connected automorphism groups characterized for toroidal horospherical varieties

Criterion established for reductivity of automorphism groups

Proved K-unstability of specific $ ext{P}^1$-bundles over rational homogeneous spaces

Abstract

We establish a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties, that is, toric fibrations over rational homogeneous spaces. The key ingredient is a characterization of the Demazure roots of toric fibers that extend to the total space. In particular, we provide a criterion for the reductivity of the connected automorphism groups of such varieties. As an application, we prove the K-unstability of certain $\mathbb{P}^1$-bundles over rational homogeneous spaces.