Chen, Huayi (2006) Positivity in algebraic geometry and in Arakelov geometry: application in algebraization and in asymptotic study of Harder-Narasimhan polygons. PhD thesis CMAT, CMAT, EP/X p.207.

Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Huayi.pdf

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.

Table of content

1. Notations et préliminaires
2. Positivité faible des fibrés vectoriels et algébrisation I: cas géométrique
3. Positivité faible des fibrés vectoriels et algébrisation II: cas arithmétique
4. Filtrations
5. Pente maximale du produit tensoriel de fibrés vectoriels hermitiens
6. Polygones de Harder-Narasimhan asymptotiques

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