Comptes Rendus
Analytic geometry
Holomorphic Cartan geometries on complex tori
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 316-321.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.02.005

Indranil Biswas 1; Sorin Dumitrescu 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2 Université Côte d'Azur, CNRS, LJAD, France
@article{CRMATH_2018__356_3_316_0,
     author = {Indranil Biswas and Sorin Dumitrescu},
     title = {Holomorphic {Cartan} geometries on complex tori},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {316--321},
     publisher = {Elsevier},
     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.02.005},
     language = {en},
}
TY  - JOUR
AU  - Indranil Biswas
AU  - Sorin Dumitrescu
TI  - Holomorphic Cartan geometries on complex tori
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 316
EP  - 321
VL  - 356
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2018.02.005
LA  - en
ID  - CRMATH_2018__356_3_316_0
ER  - 
%0 Journal Article
%A Indranil Biswas
%A Sorin Dumitrescu
%T Holomorphic Cartan geometries on complex tori
%J Comptes Rendus. Mathématique
%D 2018
%P 316-321
%V 356
%N 3
%I Elsevier
%R 10.1016/j.crma.2018.02.005
%G en
%F CRMATH_2018__356_3_316_0
Indranil Biswas; Sorin Dumitrescu. Holomorphic Cartan geometries on complex tori. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 316-321. doi : 10.1016/j.crma.2018.02.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.02.005/

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources:

Comments - Policy