Automorphisms and examples of compact non K\"ahler manifolds

TL;DR

This paper explores the properties of automorphisms on certain compact K"ahler manifolds and uses these insights to construct examples of compact non-K"ahler manifolds that challenge existing conjectures in complex geometry.

Contribution

It introduces new examples of non-K"ahler manifolds by applying automorphism properties, providing counterexamples to the abundance and Iitaka conjectures.

Findings

Automorphisms of certain K"ahler manifolds have finite order.

Constructed non-K"ahler manifolds contradict key conjectures.

Automorphism constraints lead to new manifold examples.

Abstract

If $f$ is an automorphism of a compact simply connected K\"ahler manifold with trivial canonical bundle that fixes a K\"ahler class, then the order of $f$ is finite. We apply this well known result to construct compact non-K\"ahler manifolds. These manifolds contradict the abundance and Iitaka conjectures for complex manifolds.