Wild automorphisms of compact complex spaces of lower dimensions

TL;DR

This paper classifies compact complex surfaces with wild automorphisms, showing they are either tori or certain Inoue surfaces, and explores automorphisms in higher-dimensional Kähler manifolds, revealing their properties and constraints.

Contribution

It provides a classification of wild automorphisms on compact complex surfaces and extends the understanding to higher-dimensional Kähler manifolds, generalizing previous results.

Findings

Compact complex surfaces with wild automorphisms are tori or Inoue surfaces.

Wild automorphisms on these surfaces have zero entropy.

Extended results to Kähler threefolds and fourfolds, generalizing prior work.

Abstract

An automorphism of a compact complex space is called wild in the sense of Reichstein--Rogalski--Zhang if there is no non-trivial proper invariant analytic subset. We show that a compact complex surface admitting a wild automorphism must be a complex torus or an Inoue surface of certain type, and this wild automorphism has zero entropy. As a by-product of our argument, we obtain new results about the automorphism groups of Inoue surfaces. We also study wild automorphisms of compact K\"ahler threefolds or fourfolds, and generalise the results of Oguiso--Zhang from the projective case to the K\"ahler case.