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Astérisque - Titles - 322 (2008) 151-205

Titles < 2008 < 322

Géométrie différentielle, physique mathématique, mathématiques et société (II)
Volume en l'honneur de Jean Pierre Bourguignon

Oussama Hijazi (éditeur)
Astérisque 322 (2008), xvi+256 pages
Buy the book
Presentation, Summary

Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents
Ngaiming Mok
Astérisque 322 (2008), 151-205

Résumé :
Structures géométriques sur des variétés projectives uniréglées définies par leurs variétés de tangentes rationnelles minimales
Dans un programme de recherche avec Jun-Muk Hwang nous avons étudié des structures géométriques sur les variétés projectives uniréglées, en particulier les variétés de Fano de nombres de Picard égaux à 1, definies par les variétés de tangentes rationnelles minimales associées aux espaces de modules de courbes rationnelles minimales. Dans cet article nous esquissons un dessin heuristique sur la géométrie des variétés de Fano de nombres de Picard égaux à 1 dont les variétés de tangentes rationnelles minimales sont non linéaires, en prenant comme prototypes les exemples tels ques les structures conformes holomorphes. Dans un ouvert par rapport à la topologie complexe, la structure géométrique associée aux variétés de tangentes rationnelles minimales équivaut aux données de familles de courbes holomorphes locales marquées à un point de base variable vérifiant des conditions de compatibilité. Des notions de la géometrie différentielle comme les géodésiques (nulles), la courbure et le transport parallèle constituent une source d'inspiration dans notre étude. Des formulations de problèmes suggérés par cette analogie heuristique et leurs solutions, parfois dans un contexte très générale et parfois applicables seulement aux classes de variétés de Fano spéciales, ont conduit a des résolutions d'une série de problèmes bien connus en géométrie algébrique.

Mots-clefs : Structure géométrique, courbe rationnelle minimale, variété des tangentes rationnelles minimales, application tangente, continuation analytique, caractéristique de Cauchy, courbure, prolongation, transport parallèle, faisceau tangent de nef, distribution, système différentiel, rigidité de déformation

Abstract:
In a joint research programme with Jun-Muk Hwang we have been investigating geometric structures on uniruled projective manifolds, especially Fano manifolds of Picard number 1, defined by varieties of minimal rational tangents associated to moduli spaces of minimal rational curves. In this article we outline a heuristic picture of the geometry of Fano manifolds of Picard number 1 with non-linear varieties of minimal rational tangents, taking as hints from prototypical examples such as those from holomorphic conformal structures. On an open set in the complex topology the local geometric structure associated to varieties of minimal rational tangents is equivalently given by families of local holomorphic curves marked at a variable base point satisfying certain compatibility conditions. Differential-geometric notions such as (null) geodesics, curvature and parallel transport are a source of inspiration in our study. Formulation of problems suggested by this heuristic analogy and their solutions, sometimes in a very general context and at other times applicable only to special classes of Fano manifolds, have led to resolutions of a series of well-known problems in Algebraic Geometry.

Keywords: Geometric structure, minimal rational curve, variety of minimal rational tangents, tangent map, analytic continuation, Cauchy characteristic, curvature, prolongation, parallel transport, nef tangent bundle, distribution, differential system, deformation rigidity

Class. math. : 14J45, 32M15, 32H02, 53C10


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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