Uniruled manifolds in projective and Kähler geometry, I

The goal of this lectures is to discuss some recent results on compact complex manifolds that are uniruled (i.e. covered by rational curves), but not rationally connected (i.e. two general points are not connected by a rational curve). In the first talk I will discuss a technique that aims at proving the existence of rational curves on compact Kähler manifolds, thereby generalising the work of Mori for projective manifolds. In the second talk I will explain how a manifold with nef anticanonical bundle (a generalisation of Fano manifolds) can be decomposed in its rationally connected part and a numerically trivial part. In the third talk I will discuss the technical background of both problems: the positivity of certain foliations.
I will explain all the terms appearing in this summary in the lectures. This is based on joint papers with Junyan Cao and Thomas Peternell.