Birational geometry in the study of dynamics of automorphisms and Brody/Mori/Lang hyperbolicity

TL;DR

This paper explores how birational geometry and the log minimal model program relate to the dynamics of automorphisms and hyperbolicity properties of compact Kähler manifolds, providing new insights and applications.

Contribution

It applies birational geometry and LMMP techniques to study automorphisms with positive entropy and hyperbolicity in complex algebraic geometry, offering new applications to positivity of log canonical divisors.

Findings

Relations between manifold geometry and automorphism entropy

Applications of LMMP to hyperbolic varieties

Positivity results for log canonical divisors

Abstract

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K\"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to positivity of log canonical divisor of a Mori / Brody / Lang hyperbolic (quasi-) projective (singular) variety.