Compact K\"ahler manifolds admitting large solvable groups of automorphisms

TL;DR

This paper classifies the structure of large solvable automorphism groups of compact K"ahler manifolds, extending the Tits alternative and linking group properties to the geometric nature of the manifold.

Contribution

It improves the Tits alternative for automorphism groups of K"ahler manifolds, showing their structure is either virtually abelian or related to complex tori.

Findings

G admits no non-abelian free subgroup implies specific group structure.

If G/N(G) is of rank n-1, then X is a complex torus or birational to an abelian variety.

Provides a generalization of Fujiki and Lieberman's theorem on automorphism groups.

Abstract

Let G be a group of automorphisms of a compact K\"ahler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank < n-1, or G/N(G) is a free abelian group of rank n-1 and X is a complex torus, or G is a free abelian group of rank n-1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n > 2 and is not rationally connected. We also prove and use a generalization of a theorem by Fujiki and Lieberman on the structure of Aut(X).