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Mathematics  2010 

Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves

DOI: 10.1016/j.difgeo.2009.09.003

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We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact K\"ahler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.


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