Holomorphic Cartan geometries and rational curves

Abstract

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

Keywords

• 53C55
• 53C51
• 53C56
• 53A55