Orbifolds, special varieties and classification theory

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler manifolds or Kähler manifolds with zero Kodaira dimension are special. For any X , we then construct the unique functorial fibration c X : X → C ( X ) (called its core), such that its general fibre is special, and its orbifold base is either of general type, or a point (the last case occuring if and only if X is special). We next show that the core has a canonical and functorial decomposition as a tower of fibrations with generic (orbifold) fibres either κ -rationally generated (a weak version of rational connectedness), or with zero Kodaira dimension. In particular, special manifolds are thus canonically towers of such fibrations. The main technical ingredient in the proofs is an orbifold version of Iitaka’s C n , m additivity conjecture, proved here when the orbifold base is of general type. The core of X also gives a very simple conjectural qualitative of description of both the Kobayashi pseudometric and the distribution of its K -rational points (if X is projective), description which reduces to Lang’s conjectures when X is of general type.