Hyperbolic Aspects : a Short Survey

We give a brief survey of the main results of [C04] and [C11], devoted to the bimeromorphic structure of compact Kähler manifolds X. Such manifolds are decomposed by means of iterated fibrations into elementary components, which are orbifold pairs with a canonical bundle either positive, negative, or torsion. Towers of ‘torsion and negative’ components build however the new (unconditional) class of ‘special manifolds’, which are the ones which are in a precise sense ‘opposite’ to manifolds of general type. A single funtorial (unconditional) fibration (the ‘core map’) splits any X into its two components of ‘opposite’ geometry: ‘special’ (its fibres), and general type (its orbifold base). This geometric splitting is conjectured to split X at hyperbolic and hyperbolic levels as well, leading to natural generalisations (to arbitrary smooth orbifolds (X,D)) of Lang’s conjectures, permitting to qualitatively describe in algebro-geometric terms the distribution of rational curves, rational points and entire curves on them.