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Bulletin de la SMF - Parutions - 139 (2011) 41-74

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Geometric stability of the cotangent bundle and the universal cover of a projective manifold
leavevmode unskip Fr'ed'eric Campanaand Thomas Peternell
Bulletin de la SMF 139, fascicule 1 (2011), 41-74

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Résumé :
Stabilité géométrique du fibré cotangent et du recouvrement universel d'une variété projective
Nous établissons tout d'abord un renforcement du théorème de semi-positivité de Miyaoka: le déterminant de tout quotient de toute puissance tensorielle du fibré cotangent d'une variété projective non-uniréglée est pseudo-effectif (au lieu de: génériquement nef). Une première conséquence est que est de type général si son fibré cotangent a un sous-faisceau dont le déterminant est `big'. Parmi diverses applications, nous montrons que si le rev^etement universel de n'est pas recouvert par des sous-ensembles analytiques compacts de dimension strictement positive, alors est de type général si chi (O_X)
eq 0 .Nous montrons enfin que K_X est Bbb Q -effectif si K_X+L l'est, pour un fibré en droites numériqiuement effectif sur . La démonstration de ce résultat central repose sur les travaux de C. Simpson sur les lieux de Green-Lazarsfeld, et sur les rev^etements cycliques de Viehweg. Ce résultat a été récemment étendu aux paires 'Log-canoniques' en utilisant les m^emes ingrédients.

Mots-clefs : Fibré stable, fibré en droites pseudo-effectif, dimension de Moishezon-Iitaka-`Kodaira', rev^etement universel, variété unir'eglée

Abstract:
We first prove a strengthening of Miyaoka's generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold have a pseudo-effective (instead of generically nef) determinant. A first consequence is that is of general type if its cotangent bundle contains a subsheaf with `big' determinant. Among other applications, we deduce that if the universal cover of is not covered by compact positive-dimensional analytic subsets, then is of general type if chi (O_X)
eq 0 . We finally show that if is a numerically trivial line bundle on , and if K_X+L is Bbb Q -effective, then so is K_X itself. The proof of this result rests on Simpson's work on jumping loci of numerically trivial line bundles, and Viehweg's cyclic covers. This last result is central, and has been recently extended, using the very same ingredients, to the case of log-canonical pairs.

Keywords: Bundle, pseudo-effective line bundle, Moishezon-Iitaka-`Kodaira' dimension, universal cover, uniruledness.

Class. math. : 14J40, 32Q26, 32J27, 14E30

ISSN : 0037-9484
Publié avec le concours de : Centre National de la Recherche Scientifique

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