Let $X$ be a compact connected K"ahler manifold such that the holomorphic tangent bundle $TX$ is numerically effective. A theorem of Demailly-Peternell-Schneider says that there is a finite unramified Galois covering $M o X$, a complex torus $T$, and a holomorphic surjective submersion $f: M o T$, 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 ${mathcal G}$--bundle over $T$ given by $f$, where $mathcal G$ is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle / Biswas, I.; Bruzzo, U.. - In: DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS. - ISSN 0926-2245. - 29:2(2011), pp. 147-153. [10.1016/j.difgeo.2011.02.001]

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Bruzzo, U.
2011-01-01

Abstract

Let $X$ be a compact connected K"ahler manifold such that the holomorphic tangent bundle $TX$ is numerically effective. A theorem of Demailly-Peternell-Schneider says that there is a finite unramified Galois covering $M o X$, a complex torus $T$, and a holomorphic surjective submersion $f: M o T$, 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 ${mathcal G}$--bundle over $T$ given by $f$, where $mathcal G$ is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.
2011
29
2
147
153
https://doi.org/10.1016/j.difgeo.2011.02.001
https://arxiv.org/abs/1101.4192
Biswas, I.; Bruzzo, U.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/12829
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