Some automorphism groups are linear algebraic

TL;DR

The paper investigates the structure of automorphism groups of projective varieties, proving that certain subgroups are linear algebraic and that the entire automorphism group is algebraic under specific conditions.

Contribution

It establishes that the subgroup fixing the field of invariant rational functions is linear algebraic and characterizes when the automorphism group itself is algebraic.

Findings

Subgroup fixing the invariant field is linear algebraic.

Automorphism group is algebraic if the invariant field has transcendence degree 1.

Provides structural insights into automorphism groups of projective varieties.

Abstract

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over $k$, then Aut($X$) is an algebraic group.