Invariant measures for large automorphism groups of projective surfaces

TL;DR

This paper classifies invariant measures for large automorphism groups on compact Kähler surfaces, revealing finiteness and structure of such measures, especially for K3 and Enriques surfaces, under certain dynamical conditions.

Contribution

It provides a classification of invariant measures for non-elementary automorphism groups on compact Kähler surfaces, including a complete description for K3 and Enriques surfaces.

Findings

Finiteness of invariant measures with Zariski dense support (except in Kummer cases)

Complete orbit closure description for K3 and Enriques surfaces without algebraic subsets

Classification of invariant measures under parabolic automorphisms

Abstract

We classify invariant probability measures for non-elementary groups of automorphisms, on any compact K\"ahler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.