Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Holomorphic maps from rational homogeneous spaces onto projective manifolds

Author: Chi-Hin Lau
Journal: J. Algebraic Geom. 18 (2009), 223-256
Published electronically: March 27, 2008
MathSciNet review: 2475814
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Abstract: In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if $ X$ is the image of a surjective holomorphic map from $ \mathbb{P}^n$, then $ X$ is biholomorphic to $ \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$ be a rational homogeneous manifold of Picard number $ 1$. For any surjective holomorphic map $ f:S\to 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.

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Additional Information

Chi-Hin Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Address at time of publication: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Seoul 130-012, Korea

Received by editor(s): December 13, 2006
Received by editor(s) in revised form: October 26, 2007
Published electronically: March 27, 2008

American Mathematical Society