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Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle
Authors:Indranil Biswas  Ugo Bruzzo
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
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 MX, a complex torus T, and a holomorphic surjective submersion f:MT, 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.
Keywords:32M10  14M17  53C15
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