Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana; Thomas Peternell

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 1, page 41-74
  • ISSN: 0037-9484

Abstract

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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 X have a pseudo-effective (instead of generically nef) determinant. A first consequence is that X 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 X is not covered by compact positive-dimensional analytic subsets, then X is of general type if χ ( O X ) 0 . We finally show that if L is a numerically trivial line bundle on X , and if K X + L is -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.

How to cite

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Campana, Frédéric, and Peternell, Thomas. "Geometric stability of the cotangent bundle and the universal cover of a projective manifold." Bulletin de la Société Mathématique de France 139.1 (2011): 41-74. <http://eudml.org/doc/272521>.

@article{Campana2011,
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 $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ 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 $X$ is not covered by compact positive-dimensional analytic subsets, then $X$ is of general type if $\chi (O_X)\ne 0$. We finally show that if $L$ is a numerically trivial line bundle on $X$, and if $K_X+L$ is $\mathbb \{Q\}$-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.},
author = {Campana, Frédéric, Peternell, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {bundle; pseudo-effective line bundle; Moishezon-Iitaka-‘Kodaira’ dimension; universal cover; uniruledness},
language = {eng},
number = {1},
pages = {41-74},
publisher = {Société mathématique de France},
title = {Geometric stability of the cotangent bundle and the universal cover of a projective manifold},
url = {http://eudml.org/doc/272521},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Campana, Frédéric
AU - Peternell, Thomas
TI - Geometric stability of the cotangent bundle and the universal cover of a projective manifold
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 1
SP - 41
EP - 74
AB - 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 $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ 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 $X$ is not covered by compact positive-dimensional analytic subsets, then $X$ is of general type if $\chi (O_X)\ne 0$. We finally show that if $L$ is a numerically trivial line bundle on $X$, and if $K_X+L$ is $\mathbb {Q}$-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.
LA - eng
KW - bundle; pseudo-effective line bundle; Moishezon-Iitaka-‘Kodaira’ dimension; universal cover; uniruledness
UR - http://eudml.org/doc/272521
ER -

References

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