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arXiv · 1101.4192

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Indranil Biswas·Ugo Bruzzo
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Abstract

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider 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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BibTeXRIS

Indranil Biswas, Ugo Bruzzo. 2011-01-21. Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle. https://doi.org/10.1016/j.difgeo.2011.02.001

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