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Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves (1009.5801v1)
Published 29 Sep 2010 in math.AG and math.DG
Abstract: 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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