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Connexions affines et projectives sur les surfaces complexes compactes

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Résumé

Soit (S, ∇) une surface complexe compacte connexe munie d’une connexion affine holomorphe sans torsion. Nous démontrons que ∇ est localement modelée sur une connexion affine invariante par translations sur C 2 (en particulier, ∇ est localement homogène), sauf si S est un fibré elliptique principal au-dessus d’une surface de genre g ≥ 2, de premier nombre de Betti impair et ∇ est une connexion affine holomorphe sans torsion générique sur S, auquel cas l’algèbre de Lie des champs de Killing locaux est de dimension un, engendrée par le champ fondamental de la fibration principale. Nous en déduisons que toute connexion projective holomorphe normale sur une surface complexe compacte est plate.

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  1. Mathématique, Université Paris-Sud (11), U.M.R. 8628 C.N.R.S., Bât. 425, 91405, Orsay Cedex, France

    Sorin Dumitrescu

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Dumitrescu, S. Connexions affines et projectives sur les surfaces complexes compactes. Math. Z. 264, 301–316 (2010). https://doi.org/10.1007/s00209-008-0465-8

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