On parabolic geometry, II

Abstract

Let $G$ be a simple linear algebraic group defined over $\mathbb{C}$ and $P$ a parabolic subgroup of it. Let $(M, E_P, \omega)$ be a holomorphic parabolic geometry of type $G/P$ over a smooth complex projective variety $M$. We prove that $(M, E_ , \omega)$ is holomorphically isomorphic to the standard parabolic geometry $(G/P, G, \omega_0)$ whenever $M$ is rationally connected. We then show that this is indeed the case if $M$ has Picard number one and contains a (possibly singular) rational curve. This last result is a generalization of the main result of [3], where we dealt with the case $G = PGL(d, \mathbb{C})$, $G/P = \mathbb{P}^{d-1}_{\mathbb{C}}$.