Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle |
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Authors: | Indranil Biswas Ugo Bruzzo |
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Institution: | a School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India b Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, 34013, Trieste, Italy c Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Italy |
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Abstract: | Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) 11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection. |
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Keywords: | 32M10 14M17 53C15 |
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