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Quelques problèmes de géométrie Finslérienne et Kählerienne

Abstract : This thesis deals with some classical problems in complex geometry. The first part is devoted to a problem in complex Finsler Geometry. Giving two holomorphic vector bundles E1 and E2, respectively endowed with two Finsler structures F1 and F2, we build a Finsler metric F on E 1 ⊗ E 2 involving the two initial Finsler structures. This is done under some assumptions on global sections of E1* and E2*. We give an optimal condition under which F is strictly pseudo convex with negative curvature. This result is preceded by a chapter containing a background material in complex Finsler geometry and some personal attempts. The second part of this thesis deals with a problem in Kähler Geometry. We prove the existence of an "extremal" function lower bounding all admissible functions (ie plurisubharmonic functions modulo a metric) with sup equal to zero on the complex Grassmann manifold G m,nm ( C ). The functions considered are invariant under a suitable automorphisms group. This gives a conceptually simple method to compute Tian's invariant in the case of a non toric manifold.
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Submitted on : Monday, September 7, 2015 - 10:03:28 AM
Last modification on : Thursday, December 10, 2020 - 11:04:36 AM
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Inès Adouani. Quelques problèmes de géométrie Finslérienne et Kählerienne. Géométrie métrique [math.MG]. Université Pierre et Marie Curie - Paris VI; Université Tunis El Manar. Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis (Tunisie), 2015. Français. ⟨NNT : 2015PA066130⟩. ⟨tel-01194479⟩



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