Event description
Kähler hyperbolic manifolds were introduced by Gromov around thirty years ago. In his seminal paper he showed that this class of Kähler manifolds enjoys several remarkable properties: for instance they are of general type, Kobayashi hyperbolic and their L2-Hodge (p,q)-numbers are positive if and only if p+q equals the complex dimension of the manifold. In this talk I will report about a recent joint work with S. Diverio, P. Eyssidieux and S. Trapani where we introduced a weak notion of Kähler hyperbolicity. I will explain the reasons behind our definition and I will show how some of the properties proved by Gromov for Kähler hyperbolic manifolds remain true in our more general setting. As a main application, I will describe how these ideas allow to give a positive answer to the Lang conjecture for Kähler hyperbolic manifolds.