Holomorphic maps from rational homogeneous spaces onto projective manifolds

Abstract

In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if X X is the image of a surjective holomorphic map from P n \mathbb {P}^n , then X X is biholomorphic to P n \mathbb {P}^n . This conjecture was proved by Lazarsfeld [Lect. Notes Math. 1092 (1984), 29-61] using Mori’s proof of Hartshorne’s conjecture [Ann. Math. 110 (1979), 593-606]. Then Lazarsfeld raised a more general problem, which was completely answered in the positive by Hwang and Mok. Theorem 1 ([Invent. math. 136 (1999), 209-231] and [Asian J. Math. 8 (2004), 51-63]). Let S = G / P S=G/P be a rational homogeneous manifold of Picard number 1 1 . For any surjective holomorphic map f : S → X f:S\to X to a projective manifold X X , either X X is a projective space, or f f is a biholomorphism. The aim of this article is to give a generalization of Theorem 1. We will show that modulo canonical projections, Theorem 1 is true when G G is simple without the assumption on Picard number. We need to find a dominating and generically unsplit family of rational curves which are of positive degree with respect to a given nef line bundle on X X . Such a family may not exist in general, but we will prove its existence under a certain assumption which is applicable in our situation.