no. 3
Analytic geometry
Holomorphic Cartan geometries on complex tori
[Géométries de Cartan holomorphes sur les tores complexes]

Présenté par : Jean-Pierre Demailly

Biswas, Indranil1 ; Dumitrescu, Sorin2
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2 Université Côte d'Azur, CNRS, LJAD, France
Comptes Rendus. Mathématique, Tome 356 (2018) no. 3, pp. 316-321.

Nous démontrons que, sur les tores complexes, toutes les géométries de Cartan holomorphes modelées sur (G,H), avec G groupe de Lie complexe affine, sont invariantes par translation.

In [6], it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group, on any complex torus is translation invariant.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.02.005
Biswas, Indranil 1 ; Dumitrescu, Sorin 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2 Université Côte d'Azur, CNRS, LJAD, France 
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Biswas, Indranil; Dumitrescu, Sorin. Holomorphic Cartan geometries on complex tori. Comptes Rendus. Mathématique, Tome 356 (2018) no. 3, pp. 316-321. doi : 10.1016/j.crma.2018.02.005. http://archive.numdam.org/articles/10.1016/j.crma.2018.02.005/

[1] Atiyah, M.F. Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., Volume 85 (1957), pp. 181-207

[2] Biswas, I.; Gómez, T.L. Connections and Higgs fields on a principal bundle, Ann. Glob. Anal. Geom., Volume 33 (2008), pp. 19-46

[3] Biswas, I.; McKay, B. Holomorphic Cartan geometries and Calabi–Yau manifolds, J. Geom. Phys., Volume 60 (2010), pp. 661-663

[4] Dumitrescu, S. Structures géométriques holomorphes sur les variétés complexes compactes, Ann. Sci. Éc. Norm. Supér. (4), Volume 34 (2001), pp. 557-571

[5] Dumitrescu, S. Killing fields of holomorphic Cartan geometries, Monatshefte Math., Volume 161 (2010), pp. 301-316

[6] Dumitrescu, S.; McKay, B. Symmetries of holomorphic geometric structures on complex tori, Complex Manifolds, Volume 3 (2016), pp. 1-15

[7] Ghys, É. Feuilletages holomorphes de codimension un sur les espaces homogènes complexes, Ann. Fac. Sci. Toulouse, Volume 5 (1996), pp. 493-519

[8] Gromov, M. Rigid transformation groups (Bernard, D.; Choquet-Bruhat, Y., eds.), Géométrie différentielle, Travaux en cours, vol. 33, Hermann, Paris, 1988, pp. 65-141

[9] Inoue, M.; Kobayashi, S.; Ochiai, T. Holomorphic affine connections on compact complex surfaces, J. Fac. Sci., Univ. Tokyo, Volume 27 (1980), pp. 247-264

[10] McKay, B. Holomorphic parabolic geometries and Calabi–Yau manifolds, SIGMA, Volume 7 (2011)

[11] Simpson, C.T. Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci., Volume 75 (1992), pp. 5-95

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