Orbifolds, special varieties and classification theory

Frédéric Campana[1]

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

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 3, page 499-630
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Campana, Frédéric. "Orbifolds, special varieties and classification theory." Annales de l’institut Fourier 54.3 (2004): 499-630. <http://eudml.org/doc/116120>.

@article{Campana2004,
abstract = {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 $c_X:X\rightarrow C(X)$ (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 $\kappa $-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.},
affiliation = {Université Nancy 1, département de mathématiques, BP 239, 54506 Vandoeuvre-les-Nancy (France)},
author = {Campana, Frédéric},
journal = {Annales de l’institut Fourier},
keywords = {canonical bundle; Kodaira dimension; orbifold; Kähler manifold; rational connectedness; fibration; Albanese map; Kobayashi pseudometric; rational point; Kähler manifolds; fibrations; varieties of general type; hyperbolic manifolds; arithmetic geometry; birational geometry},
language = {eng},
number = {3},
pages = {499-630},
publisher = {Association des Annales de l'Institut Fourier},
title = {Orbifolds, special varieties and classification theory},
url = {http://eudml.org/doc/116120},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Campana, Frédéric
TI - Orbifolds, special varieties and classification theory
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 3
SP - 499
EP - 630
AB - 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 $c_X:X\rightarrow C(X)$ (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 $\kappa $-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.
LA - eng
KW - canonical bundle; Kodaira dimension; orbifold; Kähler manifold; rational connectedness; fibration; Albanese map; Kobayashi pseudometric; rational point; Kähler manifolds; fibrations; varieties of general type; hyperbolic manifolds; arithmetic geometry; birational geometry
UR - http://eudml.org/doc/116120
ER -

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  7. Benoît Claudon, Andreas Höring, Compact Kähler manifolds with compactifiable universal cover
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