Métriques riemanniennes holomorphes en petite dimension  [ Holomorphic riemannian metrics in little dimension ]
Annales de l'Institut Fourier, Volume 51 (2001) no. 6, p. 1663-1690

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.

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.

DOI : https://doi.org/10.5802/aif.1870
Classification:  53B21,  53C56,  53A55
Keywords: complex manifolds, holomorphic riemannian metrics, algebraic theory of invariants, pseudogroup of local isometries
@article{AIF_2001__51_6_1663_0,
     author = {Dumitrescu, Sorin},
     title = {M\'etriques riemanniennes holomorphes en petite dimension},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {6},
     year = {2001},
     pages = {1663-1690},
     doi = {10.5802/aif.1870},
     zbl = {1016.53051},
     mrnumber = {1871285},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2001__51_6_1663_0}
}
Dumitrescu, Sorin. Métriques riemanniennes holomorphes en petite dimension. Annales de l'Institut Fourier, Volume 51 (2001) no. 6, pp. 1663-1690. doi : 10.5802/aif.1870. http://www.numdam.org/item/AIF_2001__51_6_1663_0/

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