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

Abstract

Let P be the maximal parabolic subgroup of PGL( d, C ) defined by invertible matrices ( a ij ) di,j =1 with a dj = 0 for all j ∈ [1 , d − 1]. Take a holomorphic parabolic geometry ( M, E P , ω ) of type PGL( d, C ) /P . Assume that M is a complex projective manifold. We prove the following: If there is a nonconstant holomorphic map f : CP 1 −→ M , then M is biholomorphic to the projective space CP d − 1 .