On parabolic geometry of type PGL(d,C)/P

Abstract

Let $P$ be the maximal parabolic subgroup of $\text{PGL}(d, {\mathbb C})$ defined by invertible matrices $(a_{ij})_{i,j=1}^d$ with $a_{dj}\,=\, 0$ for all $j\, \in\, [1\, ,d-1]$. Take a holomorphic parabolic geometry $(M\, ,E_P\, ,\omega)$ of type $\text{PGL}(d,{\mathbb C})/P$. Assume that $M$ is a complex projective manifold. We prove the following: If there is a nonconstant holomorphic map $f\, :\, {\mathbb C} {\mathbb P}^1\,\longrightarrow \, M$, then $M$ is biholomorphic to the projective space ${\mathbb C}{\mathbb P}^{d-1}$.