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Sorin Dumitrescu
Homogénéité locale pour les métriques riemanniennes holomorphes en dimension $3$
(Holomorphic riemannian metrics on compact threefolds are locally homogeneous)
Annales de l'institut Fourier, 57 no. 3 (2007), p. 739-773, doi: 10.5802/aif.2275
Article PDF | Reviews MR 2336828 | Zbl 1128.53045
Class. Math.: 53B21, 53C56, 53A55
Keywords: complex manifolds, holomorphic riemannian metrics, rigid structures, pseudogroup of local isometries

Résumé - Abstract

A holomorphic Riemannian metric on a compact complex manifold $M$ is a holomorphic section $q$ of the bundle $S^2(T^{*}M)$ of complex quadratic forms on the holomorphic tangent bundle on $M$ such that $q(m)$ is non degenerated (of maximal rank) for each point $m$ in $M$. This is an analogous of a (real) riemannian metric in the setting of the complex geometry. Contrary to the situation in the real framework, few complex compact manifolds admit holomorphic riemannian metrics. In this paper we show that any holomorphic riemannian metric on a compact complex connected threefold is locally homogeneous ( i.e. the pseudogroup of local isometries acts transitively on $M$). In some particular situations, this leads to classification results. Our method is a mixture of analytic geometry, invariant theory for algebraic actions and differential geometry of Gromov’s rigid geometric structures.

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