Compact Kähler varieties with pseudo-effective tangent sheaf and a dynamical application

Positivity properties of tangent sheaves are known to impose strong restrictions on the global geometry of projective varieties and, more generally, compact Kähler varieties. In this talk, after briefly reviewing the background, we will present several structural results for compact Kähler varieties with pseudo-effective tangent sheaf. As an application to complex dynamics, we will also establish a structure theorem for compact Kähler manifolds admitting a surjective holomorphic self-map with dominant topological degree. The talk is based on joint works with Shin-ichi Matsumura.