# Déformation de variétés kählériennes compactes : invariance de la $\Gamma$-dimension et extension de sections pluricanoniques

Abstract : In this thesis we study universal cover of Kähler compact manifolds, their pluricanonical systems and the different links between them. First, we introduce the $\Gamma$-reduction of a Kähler compact manifold as a rational Remmert reduction of its universal cover ; the $\Gamma$-dimension is defined to be the dimension of the base of this fibration. In this study we consider the following aspects : behaviour of the $\Gamma$-dimension in a fibration, relationships with $L^2$ holomorphic forms on the universal cover, comparison with the fibrations of the classification theory, $\Gamma$-reduction for manifolds of small dimension. At the end of this first part, we establish invariance of $\Gamma$-dimension for several families of Kähler threefolds (for instance for non general type). We then show statements of extension of pluricanonical forms in the spirit of the One-Tower method. After a brief review concerning positivity of line bundles and multiplier ideal sheaves, we apply this strategy in different situations : projective family (with a twisting pseudo-effective line bundle), hypersurface in a projective manifold, $\Gamma$-reduction for manifolds of general type and family of infinite covers.
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https://tel.archives-ouvertes.fr/tel-01748277
Contributor : Benoît Claudon <>
Submitted on : Wednesday, December 19, 2007 - 10:17:46 AM
Last modification on : Friday, February 26, 2021 - 3:22:10 AM
Long-term archiving on: : Monday, April 12, 2010 - 8:27:18 AM

### Identifiers

• HAL Id : tel-01748277, version 2

### Citation

Benoît Claudon. Déformation de variétés kählériennes compactes : invariance de la $\Gamma$-dimension et extension de sections pluricanoniques. Mathématiques [math]. Université Henri Poincaré - Nancy I, 2007. Français. ⟨NNT : 2007NAN10093⟩. ⟨tel-01748277v2⟩

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