%0 Journal Article %T Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves %A Indranil Biswas %A Benjamin McKay %J Mathematics %D 2010 %I arXiv %R 10.1016/j.difgeo.2009.09.003 %X 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. %U http://arxiv.org/abs/1009.5801v1