Positivité en géométrie algébrique et en géométrie d'Arakelov :
application à l'algébrisation et à l'étude asymptotique des polygones de
Harder-Narasimhan

Abstract : The objective of this thesis is to study various concepts of positivity, in algebraic geometry and in Arakelov geometry, for vector bundles on a projective variety, and to develop applications to the study of the algebraicity of formal sub-schemes of algebraic varieties and to the asymptotic study of Harder-Narasimhan polygons.

In the first part of the thesis, we propose a condition called P3 of a vector bundle on a projective variety of dimension at least 1. We show that this condition is weaker than the amplitude of the vector bundle, and in the context of complex algebraic geometry, is weaker that 1-positivity. We then show that the P3 condition for the normal bundle of the scheme of definition in the sub-formal scheme suffits to ensure the algebraicity of the sub-formal scheme. Finally, we apply this algebraicity criterion on the comparison of equivalence in a etale neighbourhood and that in a formal neighbourhood of two pairs of schemes. We also discuss the analogue of the P3 condition in the context of Arakelov geometry.

In the second part of this thesis, we propose a new point of view of the Harder-Narasimhan filtration of a vector bundle (resp. hermitian vector bundle) on a smooth projective curve (resp. the spectrum of a algebraic integer ring). With this point of view, in stead of study directly the Harder-Narasimhan filtration or polygon, we may study the associated Borel measure on R. Combing this interpretation with an combinatary argument, we show that, under some weak technical conditions, the (normalized) Harder-Narasimhan polygons associated to a graded algebra of finite type in (hermitian) vector bundles converge uniformly to a concave curve on [0,1], where the proof of the arithmetic part uses a new estimation of the maximum slope of the tensor product of several hermitian vector bundles which is developped in this thesis.
Document type :
Theses
Complete list of metadatas

Cited literature [96 references]  Display  Hide  Download

https://pastel.archives-ouvertes.fr/tel-00119162
Contributor : Huayi Chen <>
Submitted on : Wednesday, January 24, 2007 - 5:15:50 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
Long-term archiving on : Tuesday, September 21, 2010 - 11:50:59 AM

Files

Identifiers

  • HAL Id : tel-00119162, version 2

Collections

Citation

Huayi Chen. Positivité en géométrie algébrique et en géométrie d'Arakelov :
application à l'algébrisation et à l'étude asymptotique des polygones de
Harder-Narasimhan. Mathématiques [math]. Ecole Polytechnique X, 2006. Français. ⟨tel-00119162v2⟩

Share

Metrics

Record views

276

Files downloads

233