Orbifolds, special varieties and classification theory

Campana, Frédéric

doi:10.5802/aif.2027

Orbifolds, special varieties and classification theory
[Orbifoldes, variétés spéciales et théorie de la classification]

Campana, Frédéric ¹

¹ Université Nancy 1, département de mathématiques, BP 239, 54506 Vandoeuvre-les-Nancy (France)

Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 499-630.

Résumé
Abstract

Le présent article décrit à l’aide de fibrations fonctorielles intrinsèques, la structure géométrique (et conjecturalement la pseudométrique de Kobayashi ainsi que l’arithmétique dans le cas projectif) des variétés Kählériennes compactes. Les variétés spéciales sont tout d’abord définies comme étant les variétés Kählériennes compactes ne possédant pas d’application méromorphe surjective sur une orbifolde de type général, la structure d’orbifolde de la base provenant du diviseur des fibres multiples. On montre que les variétés Kählériennes compactes qui sont, soit rationnellement connexes, soit à dimension de Kodaira nulle sont spéciales. Nous construisons ensuite fonctoriellement, pour toute telle variété $X$ , l’ unique fibration $c_{X} : X \to C (X)$ (le coeur de $X$ ) dont les fibres sont spéciales et dont la base orbifolde est soit de type général, soit un point (ce dernier cas se produisant si et seulement si $X$ est spéciale). Le coeur de $X$ est ensuite canoniquement décomposé comme une tour de fibrations à fibres soit $κ$ -rationnellement engendrées (une version faible de la connéxité rationnelle), soit à dimension de Kodaira nulle. En particulier, les variétés spéciales sont donc de telles tours de fibrations. L’ingrédient technique essentiel des démonstrations est une version orbifolde de la conjecture $C_{n, m}$ d’Iitaka, établie lorsque la base orbifolde est de type général. Le coeur de $X$ permet de donner une description qualitative conjecturale très simple de la pseudométrique de Kobayashi de $X$ et de la distribution de ses points $K$ -rationnels (si $X$ est projective, définie sur le corps $K$ de type fini sur $ℚ$ ), description se réduisant à celle de Lang lorsque $X$ est de type général.

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 $c_{X} : X \to C (X)$ (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.

EuDML MR Zbl

DOI : 10.5802/aif.2027

Classification : 14C30, 14D10, 14E05, 14G05, 14J40, 32J27, 32Q15, 32Q57
Keywords: canonical bundle, Kodaira dimension, orbifold, Kähler manifold, rational connectedness, fibration, Albanese map, Kobayashi pseudometric, rational point.
Mot clés : fibré canonique, dimension de Kodaira, orbifolde, variété Kählérienne compacte, connexité rationnelle, fibration, morphisme d'Albanese, pseudométrique de Kobayashi, points rationnels.

Affiliations des auteurs :

Campana, Frédéric ¹

¹ Université Nancy 1, département de mathématiques, BP 239, 54506 Vandoeuvre-les-Nancy (France)

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     title = {Orbifolds, special varieties and classification theory},
     journal = {Annales de l'Institut Fourier},
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Campana, Frédéric. Orbifolds, special varieties and classification theory. Annales de l'Institut Fourier, Tome 54 (2004) no. 3, pp. 499-630. doi : 10.5802/aif.2027. https://aif.centre-mersenne.org/articles/10.5802/aif.2027/

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