On automorphisms and endomorphisms of projective varieties

TL;DR

This paper explores the structure of automorphism groups of projective varieties and characterizes which algebraic semigroups can be realized as endomorphisms of such varieties, revealing structural limitations.

Contribution

It demonstrates that any connected algebraic group over a perfect field can be realized as the automorphism group of a normal projective variety and characterizes the realizability of algebraic semigroups as endomorphisms.

Findings

Connected algebraic groups are realizable as automorphism groups of some normal projective variety.

Few connected algebraic semigroups can be realized as endomorphisms of a projective variety.

Structural description of connected subsemigroup schemes of End(X).

Abstract

We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X).