On Ramanujam's Theorem About Finite Dimensional Groups of Automorphisms

TL;DR

This paper extends Ramanujam's theorem, which states that connected finite-dimensional subgroups of automorphism groups of irreducible varieties are algebraic, to arbitrary varieties, clarifying the notion of dimension.

Contribution

It generalizes Ramanujam's theorem to non-irreducible varieties and discusses the concept of dimension in this broader context.

Findings

Connected finite-dimensional automorphism subgroups are algebraic for irreducible varieties.

Extension of the theorem to arbitrary varieties.

Clarification of the notion of dimension for automorphism groups.

Abstract

Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam's theorem to arbitrary (not necessarily irreducible) varieties.