Abstract
Let G be a connected complex Lie group and \({\Gamma \subset G}\) a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E H over G/Γ admits a holomorphic connection if and only if E H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E H over G/Γ admits a flat holomorphic connection if and only if E H is homogeneous.
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References
Atiyah M.F.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc 85, 181–207 (1957)
Becker K. et al.: Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory. Nuclear Phys B 751, 108–128 (2006)
Biswas I., Gómez T.L.: Connections and Higgs fields on a principal bundle. Ann. Glob. Anal. Geom 33, 19–46 (2008)
N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. 1285, Hermann, Paris, 1960.
Fernandez M. et al.: Non-Kähler Heterotic String Compactifications with non-zero fluxes and constant dilaton. Comm. Math. Phys 288, 677–697 (2009)
Goldstein E., Prokushkin S.: Geometric model for complex non-Kähler manifolds with SU(3) structure. Comm. Math. Phys 251, 65–78 (2004)
Grantcharov G.: Geometry of compact complex homogeneous spaces with vanishing first Chern class. Adv. Math 226, 3136–3159 (2011)
Grantcharov D., Grantcharov G., Poon Y.: Calabi-Yau connections with torsion on toric bundles. J. Diff. Geom 78, 13–32 (2008)
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Biswas, I. Principal bundles on compact complex manifolds with trivial tangent bundle. Arch. Math. 96, 409–416 (2011). https://doi.org/10.1007/s00013-011-0268-8
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DOI: https://doi.org/10.1007/s00013-011-0268-8