Automorphism groups of almost homogeneous varieties

TL;DR

This paper investigates the structure of automorphism groups of almost homogeneous varieties, establishing conditions under which these groups are linear algebraic or arithmetic, and providing criteria for when a group can be the full automorphism group.

Contribution

It proves that Aut(X) is a linear algebraic group if G is, and that the component group of Aut(X) is arithmetic for arbitrary G, also giving conditions for G to be the full automorphism group.

Findings

Aut(X) is linear algebraic if G is linear algebraic.

The component group of Aut(X) is arithmetic for any G.

Provides conditions for G to be the full automorphism group of a variety.

Abstract

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is arithmetic. Along the way, we obtain a restrictive condition for $G$ to be the full automorphism group of some normal projective variety.