On some problems of Kobayashi and Lang; algebraic approaches

Abstract

The goal of this survey paper is to present results on hyperbolicity of complex algebraic manifolds, which appeared after the papers [48] and [42], where a number of important and intriguing conjectures were proposed. Since the paper puts the accent on measure hyperbolicity on one hand, and on algebraic methods on the other hands, we hope it will not overlap too much with the beautiful paper [21] by Demailly, except for the basic definitions and starting points. The basic questions asked in [42], and in a different spirit in [48] concern the relationships between curvature properties of a given complex manifold (or complex algebraic variety) on one hand, and the behaviour of holomorphic maps from disks or polydisks to them, on the other hand, or, from a more algebraic point of view, the (non) existence of subvarieties of certain types.