Automorphism groups of positive entropy on projective threefolds

TL;DR

This paper investigates the structure of automorphism groups with positive entropy on projective threefolds, revealing their algebraic properties and conditions under which the threefolds are complex tori.

Contribution

It establishes a connection between automorphism groups of positive entropy and semi-simple algebraic groups, and characterizes when threefolds are complex tori based on group actions.

Findings

Automorphism groups modulo null entropy are Zariski-dense in semi-simple real algebraic groups.

If the automorphism group is almost abelian with positive rank and the kernel is infinite, the threefold is a complex torus.

The results apply to minimal threefolds and relate group actions to geometric structures.

Abstract

We prove two results about the natural representation of a group G of automorphisms of a normal projective threefold X on its second cohomology. We show that if X is minimal then G, modulo a normal subgroup of null entropy, is embedded as a Zariski-dense subset in a semi-simple real linear algebraic group of real rank < 3. Next, we show that X is a complex torus if the image of G is an almost abelian group of positive rank and the kernel is infinite, unless X is equivariantly non-trivially fibred.