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Stabilité et filtration de Harder-Narasimhan

Abstract : Introduced on algebraic manifolds, the notion of stability has been generalized to the case of Kähler manifolds and then to any compact complex manifold using Gauduchon's metrics. The behavior of non semi-stable fiber bundles (or coherent sheaves) had only been studied in the algebraic case and had been described by means of the notion of Harder-Narasimhan filtration (HNF). In the present work, we carry on with this study for any compact complex (possibly non Kähler) manifold. For any complex vector bundle, we prove the existence of a subsheaf of maximal degree. This subsheaf arises as a limit in the sense of ``weakly holomorphic subbundles''. This notion was first introduced by Uhlenbeck and Yau in their study of the Kobayashi-Hitchin correspondence and actually provides us with ``the good notion'' of convergence. In this context, we prove the existence of a HNF. We then generalize these results to the case of torsion-free coherent sheaves, which leads to important convergence questions resulting from the non compactness of the basis (the set where the sheaf is locally free). We also show how to apply these methods to families of fiber bundles (or flat families of torsion-free sheaves) over a deformation of compact complex manifolds to get existence theorems for limit subsheaves similar to Bishop's theorem. By the same, we get a new proof of the openness property of the stability in deformation. This proof does not use the difficult Kobayashi-Hitchin correspondance. In a second part, we study simplicity and stability conditions for the tangent bundle of a compact complex surface of the class $VII$. In particular, we obtain an example of deformation of a surface with global spherical shell illustrating the non openness of the non semi-stability property in deformation.
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https://tel.archives-ouvertes.fr/tel-00004129
Contributor : Laurent Bruasse <>
Submitted on : Friday, January 9, 2004 - 10:11:12 AM
Last modification on : Thursday, October 11, 2018 - 1:19:41 AM
Long-term archiving on: : Friday, April 2, 2010 - 7:58:06 PM

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Laurent Bruasse. Stabilité et filtration de Harder-Narasimhan. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2001. Français. ⟨tel-00004129⟩

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