Structures affines et projectives sur les surfaces complexes

Klingler, Bruno

Annales de l'Institut Fourier, Tome 48 (1998), p. 441-477 / Harvested from Numdam

Résumé

Une structure complexe affine (resp. projective) sur une surface complexe est la donnée d’un atlas de cartes à valeur dans ${\Bbb C}^{2}$ (resp. ${\bf P}^{2}{\Bbb C}$) à changements de cartes localement constants dans le groupe affine $A(2,{\Bbb C})$ (resp. le groupe ${\bf P}GL(3,{\Bbb C})$). Dans cet article nous classifions les surfaces complexes affines et calculons, à surface complexe $S$ fixée, l’espace de déformation des structures complexes affines sur $S$ compatibles avec sa structure analytique. Nous montrons aussi que toute structure projective sur une surface complexe admettant une structure complexe affine est nécessairement affine.

Abstract

A complex affine (resp. projective) structure on a complex surface is an atlas of charts with value in ${\Bbb C}^{2}$ (resp. in ${\bf P}^{2}{\Bbb C}$) with change of coordinates in the affine group $A(2,{\Bbb C})$ (resp. ${\bf P}GL(3,{\Bbb C})$). In this paper we classify the affine complex surfaces and calculate, given a complex surface $S$, the space of deformation of affine structures on $S$ compatible with its analytic structure. We also show that a projective structure on a complex surface admitting an affine structure is necessarily affine.

Détail

Publié le : 1998-01-01

MR 99c:32038 | Zbl 0920.32027

DOI : https://doi.org/10.5802/aif.1625

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