Ja n 20 02 SPECIAL VARIETIES AND CLASSIFICATION THEORY

A new class of compact Kähler manifolds, called special, is defined: they are the ones having no surjective meromorphic map to an orbifold of general type. The special manifolds are in many respect higher-dimensional generalisations of rational and elliptic curves. For example, we show that being rationally connected or having vanishing Kodaira dimension implies being special. For any compact Kähler X, we further construct a fibration c X : X → C(X), which we call its core, such that the general fibres of c X are special, and every special subvariety of X containing a general point of X is contained in the corresponding fibre of c X. We then conjecture and prove in low dimensions and some other cases that: 1) Special manifolds have an almost abelian fundamental group. 2) Special manifolds are exactly the ones having a vanishing Kobayashi pseudometric. 3) Projective special manifolds are exactly the ones having a " potentially dense " set of K-rational points, if defined over the field K ⊂ C. (See definition in section 7). 4) The core is a fibration of general type, which means that so is its base C(X), when equipped with its orbifold structure coming from the multiple fibres of c X. 5) The Kobayashi pseudometric of X is obtained as the pull-back of the orbifold Kobayashi pseudo-metric on C(X), which is a metric outside some proper algebraic subset. 6) If X is projective, defined over some finitely generated (over Q) subfield K of the complex number field, the set of K-rational points of X is mapped by the core into a proper algebraic subset of C(X). These two last conjectures are the natural generalisations for arbitrary X of Lang's conjectures formulated when X is of general type. §0.Introduction (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), hy-perbolicity 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 its " special part " …