Structures affines et projectives sur les surfaces complexes

Klingler, Bruno

Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 441-477.

Une structure complexe affine (resp. projective) sur une surface complexe est la donnée d’un atlas de cartes à valeur dans ${ℂ}^2$ (resp. ${\mathbf{P}}^2ℂ$) à changements de cartes localement constants dans le groupe affine $A(2,ℂ)$ (resp. le groupe $\mathbf{P}GL(3,ℂ)$). 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.

A complex affine (resp. projective) structure on a complex surface is an atlas of charts with value in ${ℂ}^2$ (resp. in ${\mathbf{P}}^2ℂ$) with change of coordinates in the affine group $A(2,ℂ)$ (resp. $\mathbf{P}GL(3,ℂ)$). 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.

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