Special Varieties and Classification Theory: An Overview

Lang's conjectures link the geometric, hyperbolic, and arithmetic properties of projective complex varieties of general type. We propose here an extension of these conjectures to arbitrary projective varieties X. This extension rests on the notion of ‘special’ variety. This class contains manifolds either rationally connected or with vanishing Kodaira dimension. We further construct for any Xits ‘core’, which is a fibration cX: X→C(X) with general fibre special and orbifold base of general type. This fibration seems to permit us to decompose Xaccording to the dichotomy ‘special’ vs ‘general type’, and not only leads to the above-mentioned extension of Lang's conjectures but also to a simple global view of classification theory.