Automorphism groups of toroidal horospherical varieties
Lorenzo Barban, DongSeon Hwang, Minseong Kwon
TL;DR
The paper addresses the structure of the connected automorphism group $\mathrm{Aut}^0(X)$ for smooth complete toroidal horospherical varieties, reducing the problem to toric fibers over rational homogeneous bases. It introduces an extension of Demazure roots to the horospherical setting and uses Blanchard’s lemma and representation-theoretic tools to derive a Levi decomposition and a concrete reductivity criterion: $\mathrm{Aut}^0(X)$ is reductive iff the unipotent Demazure-root set ${\mathcal U}^+_{G}(X)$ is empty. The results yield a precise Levi-subgroup description generated by $\mathrm{Aut}_G(X)$ and certain $U_m^+$-subgroups, and they apply to projective bundles $X=\mathbb P_Y(\bigoplus_i L_i)$ over rational homogeneous spaces, giving explicit reductivity and K-stability criteria. As applications, the authors provide explicit constructions of K-unstable Fano $\mathbb P^1$-bundles and criteria that relate nefness of line-bundle differences to automorphism-group reductivity, advancing understanding of stability phenomena in horospherical and toric-fibered geometries.
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. A key ingredient is an extension of the notion of Demazure roots from toric varieties to toroidal horospherical varieties. 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.