In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if X is the image of a surjective holomorphic map from P-n, then X is biholomorphic to 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 be a rational homogeneous manifold of Picard number 1. For any surjective holomorphic map f : S -> X to a projective manifold X, either X is a projective space, or 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 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. Such a family may not exist in general, but we will prove its existence under a certain assumption which is applicable in our situation.
展开▼
机译：在[数学。安142，453-468]，Remmert和Van de Ven推测，如果X是来自P-n的射影全同图的图像，则X是P-n的全纯。拉扎斯菲尔德[Lect。笔记数学。 1092（1984），29-61]使用森对Hartshorne猜想的证明[Ann。数学。 110（1979），593-606]。然后，Lazarsfeld提出了一个更普遍的问题，Hwang和Mok的肯定回答完全。定理1（[Invent。math.136（1999），209-231]和[Asian J. Math。8（2004），51-63]）。令S = G / P是皮卡德数1的有理齐次流形。对于任何射影同胚图f：S-> X到射影流形X，X是射影空间，或者f是双全纯。本文的目的是给出定理1的一般化。我们将证明模正则投影，当G不为Picard数假设时，当G为简单时，定理1为真。我们需要找到一个相对于X上给定的nef线束具有正度的，占主导地位且通常不分裂的有理曲线族。这种族通常可能不存在，但是我们将在一定假设下证明其存在，即适用于我们的情况。
展开▼
