Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Details Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

2011-01-21

arxiv.org/abs/1101.4192v1

Diff. Geom. Appl. 29 (2011) 147-153

Mathematics

Complex Variables

11 pages. To be published in Diff. Geom. Appl

Scientific paper

10.1016/j.difgeo.2011.02.001

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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