Cohomologically hyperbolic endomorphisms of complex manifolds

TL;DR

This paper proves that compact Kahler manifolds with cohomologically hyperbolic surjective endomorphisms have non-positive Kodaira dimension and characterizes their geometric structure and fundamental groups in dimension 3.

Contribution

It confirms Guedj's conjecture in the holomorphic case and classifies the structure of such manifolds in dimension three.

Findings

Kodaira dimension is non-positive for these manifolds.

Provides classification of geometric structures in dimension three.

Determines fundamental groups up to finite index.

Abstract

We show that if a compact Kahler manifold X admits a cohomologically hyperbolic surjective endomorphism then its Kodaira dimension is non-positive. This gives an affirmative answer to a conjecture of Guedj in the holomorphic case. The main part of the paper is to determine the geometric structure and the fundamental groups (up to finite index) for those X of dimension 3.