Automorphisms and examples of compact non K\"ahler manifolds

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.