Special Varieties and Classification Theory

(0.0) For projective curves,there exists a fundamental dichotomy between curves of genus 0 or 1 on one side, and curves of genus 2 or more on the other side.This dichotomy appears at many levels, such as: Kodaira dimension, topology (fundamental group), hyperbolicity properties (as expressed by the Kobayashi pseudo-metric), and arithmetic geometry (see [La 1,2] and section 7 below). The objective of this article is to introduce a natural generalisation of this dichotomy for higher-dimensional Kähler manifolds, and to show how to decompose intrinsically and functorially, by means of a single fibration, which we call the core, any compact Kähler manifold into manifolds belonging to one of the two classes of our generalised dichotomy. Our first class consists of special manifolds, generalising curves of genus 0 or 1; the second class is that of orbifolds of general type (the orbifold structure unfortunately seems unavoidable), generalising curves of genus 2 or more. The special manifolds are defined as the ones having no surjective map to an orbifold of general type, taking care of the multiple fibres (which provide the target with the orbifold structure). The manifolds either rationally connected, or having Kodaira dimension zero are shown to be special, as expected, since they naturally generalise curves with genus 0 or 1. We moreover conjecture that, just as for curves, special manifolds are also exactly the manifolds with vanishing Kobayashi pseudo-metric (Conjecture III), have virtually abelian fundamental groups (Conjecture I), and form a class which is stable under deformation. We show these properties in special cases and in dimension up to three in most cases. Furthermore, for each X we construct a canonical fibration c X : X→C(X) (we call it its core) with its general fibre being special and maximal for this property. We then conjecture (and prove in dimensions up to three) that its orbifold base is of general type (Conjecture II). This decomposition thus also reduces by Conjecture III the Kobayashi pseudometric of X to a pseudometric coming from the orbifold base, since it should vanish along the fibres. Since our base orbifold should be of general type by Conjecture II, we are naturally lead to formulate extensions of Lang's conjectures: our Conjecture IV states that the orbifold pseudometric on C(X) which gives that of X by pullback under c X is a metric outside a proper algebraic subset S of C(X). On the arithmetic side, if X is …