no. 6
Métriques riemanniennes holomorphes en petite dimension
Dumitrescu, Sorin1
1 Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay cedex (France)
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.

DOI : 10.5802/aif.1870
Classification : 53B21, 53C56, 53A55
Mot clés : variétés complexes, métriques riemanniennes holomorphes, théorie algébrique des invariants, pseudo-groupe d'isométries locales
Keywords: complex manifolds, holomorphic riemannian metrics, algebraic theory of invariants, pseudogroup of local isometries

Dumitrescu, Sorin 1

1 Université Paris-Sud, Mathématiques, Bâtiment 425, 91405 Orsay cedex (France) 
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Dumitrescu, Sorin. Métriques riemanniennes holomorphes en petite dimension. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1663-1690. doi : 10.5802/aif.1870. https://aif.centre-mersenne.org/articles/10.5802/aif.1870/

[1] D. Alekseevskij; A. Vinogradov; V. Lychagin; E.M.S Basic ideas and concepts of differential geometry, Geometry I, E.M.S., Volume 28 (1991)

[2] W. Barth; C. Peters; A. Van de Ven Compact complex surfaces, Springer-Verlag, 1984 | MR | Zbl

[3] Y. Benoist Orbites des structures rigides (d'après M. Gromov), Feuilletages et systèmes intégrables (Montpellier, 1995) (1997), pp. 1-17 | Zbl

[4] F. Bogomolov Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izvestija, Volume 13 (1979) no. 3, pp. 499-555 | DOI | Zbl

[5] M. Brion Sur l'image de l'application moment, Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986) (1987), pp. 177-192 | Zbl

[6] M. Brunella On holomorphic forms on compact complex threefolds, Comment. Math. Helv., Volume 74 (1999) no. 4, pp. 642-656 | DOI | MR | Zbl

[7] G. D'Ambra; M. Gromov Lectures on transformations groups: geometry and dynamics, Surveys in Differential Geometry (Cambridge) (1990), pp. 19-111 | MR | Zbl

[8] S. Dumitrescu Structures géométriques holomorphes (1999) (Thèse E.N.S.-Lyon) | MR | Zbl

[9] S. Dumitrescu Structures géométriques holomorphes sur les variétés complexes compactes (à paraître aux Annales Scientifiques de l'E.N.S) | Numdam | Zbl

[10] I. Enoki; eds. Mabuchi et al. Generalizations of Albanese mappings for non-Kähler manifolds, Geometry and analysis on complex manifolds (1994), pp. 51-62 | Zbl

[11] M. Gromov Rigid transformation groups, Géométrie Différentielle, Travaux en cours, Volume 33 (1988), pp. 65-141 | Zbl

[12] E. Ghys Déformations des structures complexes sur les espaces homogènes de S L ( 2 , ℂ ) , J. reine angew. Math., Volume 468 (1995), pp. 113-138 | DOI | MR | Zbl

[13] P. Griffiths; J. Harris Principles of algebraic geometry, Wiley-interscience publication, 1978 | MR | Zbl

[14] J. Humphreys Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer-Verlag, 1975 | Zbl

[15] M. Inoue; S. Kobayashi; T. Ochiai Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Volume 27 (1980) no. 2, pp. 247-264 | MR | Zbl

[16] S. Kobayashi The first Chern class and holomorphic symmetric tensor fields, J. Math. Soc. Japan, Volume 32 (1980) no. 2, pp. 325-329 | DOI | MR | Zbl

[17] B. Moishezon On n dimensional compact varieties with n independent meromorphic functions, Amer. Math. Soc. Transl., Volume 63 (1967), pp. 51-77 | Zbl

[18] D. Mumford Introduction to algebraic geometry (1966)

[19] K. Nomizu On local and global existence of Killing vector fields, Ann. of Math. (2), Volume 72 (1960), pp. 105-120 | DOI | MR | Zbl

[20] I. Singer Infinitesimally homogeneous spaces, Comm. Pure Appl. Math, Volume 13 (1960), pp. 685-697 | DOI | MR | Zbl

[21] K. Ueno Classification theory of algebraic varieties and compact complet spaces, Springer Lect. Notes, Volume 439 (1975) | MR | Zbl

[22] C. Wall Geometric structures on compact complex analytic surfaces, Topology, Volume 25 (1986) no. 2, pp. 119-153 | DOI | MR | Zbl

[23] J. Wolf Spaces of constant curvature, McGraw-Hill Series in Higher Math. (1967) | MR | Zbl

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