Métriques riemanniennes holomorphes en petite dimension

Dumitrescu, Sorin

Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1663-1690.

Nous étudions les métriques riemanniennes holomorphes sur les variétés complexes compactes de dimension $3$. Nous montrons que, contrairement au cas réel, une métrique riemannienne holomorphe possède un “grand” pseudo-groupe d’isométries locales. Ceci implique qu’une telle métrique n’existe pas sur les variétés complexes compactes simplement connexes de dimension $3$.

We study holomorphic Riemannian metrics on compact complex threefolds. We show that, contrary to the situation in the real domain, a holomorphic Riemannian metric admits a "big" pseudogroup of local isometries. It follows that compact complex simply connected threefolds do not admit any holomorphic Riemannian metric.

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