Appears in collection : Conférence à la mémoire de Jean-Pierre Demailly

In his wonderful book "Shafarevich maps and automorphic forms", J. Kollár asked almost 30 years ago for a "good" birational version of the notion of Kähler hyperbolicity introduced by M. Gromov in the early '90s. Kähler hyperbolic manifolds are compact Kähler manifold admitting a Kähler form whose pull-back to the universal cover becomes d-exact and moreover with a bounded primitive. Such manifolds are of general type (more than this: with ample canonical bundle), Kobayashi hyperbolic, and with large fundamental group. We shall report on the recently intruduced class of weakly Kähler hyperbolic manifolds which indeed provides such a birational generalization. Among other things, we shall explain that these manifolds are of general type, satisfy a precise quantitative version of the Green-Griffiths conjecture and have generically arbitrarily large fundamental group. Moreover, their properties permit to verify Lang's conjecture for Kähler hyperbolic manifolds. This is a joint work with F. Bei, B. Claudon, P. Eyssidieux, and S. Trapani.