Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number $ 1$


Authors: Jaehyun Hong and Ngaiming Mok
Journal: J. Algebraic Geom. 22 (2013), 333-362
DOI: https://doi.org/10.1090/S1056-3911-2012-00611-0
Published electronically: November 5, 2012
MathSciNet review: 3019452
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Abstract | References | Additional Information

Abstract: In a series of works one of the authors has developed with J.-M. Hwang a geometric theory of uniruled projective manifolds basing on the study of varieties of minimal rational tangents, and the geometric theory has especially been applied to rational homogeneous manifolds of Picard number 1. In Mok [Astérisque 322, pp. 151-205] and Hong-Mok [J. Diff. Geom. 86 (2010), pp. 539-567] the authors have started the study of uniruled projective subvarieties, and a method was developed for characterizing certain subvarieties of rational homogeneous manifolds. The method relies on non-equidimensional Cartan-Fubini extension and a notion of parallel transport of varieties of minimal rational tangents.

In the current article we apply the notion of parallel transport to a characterization of smooth Schubert varieties of rational homogeneous manifolds of Picard number 1. Given a pair $ (S, S_0)$ consisting of a rational homogeneous manifold $ S$ of Picard number 1 and a smooth Schubert variety $ S_0$ of $ S$, where no restrictions are placed on $ S_0$ when $ S = G/P$ is associated to a long root (while necessarily some cases have to be excluded when $ S$ is associated to a short root), we prove that any subvariety of $ S$ having the same homology class as $ S_0$ must be $ gS_0$ for some $ g \in$$ \text {Aut}(S)$.

We reduce the problem first of all to a characterization of local deformations $ S_t$ of $ S_0$ as a subvariety of $ S$. By Kodaira stability, $ S_t $ is uniruled by minimal rational curves of $ S$ lying on $ S_t$. We establish a biholomorphism between $ S_t$ and $ S_0$ which extends to a global automorphism by reconstructing $ S_t$ by means of a repeated use of parallel transport of varieties of minimal rational tangents along minimal rational curves issuing from a general base point. Our method is applicable also to the case of singular Schubert varieties provided that there exists a minimal rational curve on the smooth locus of the variety.


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Additional Information

Jaehyun Hong
Affiliation: Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, 599 Gwanak-ro Gwanak-gu Seoul 151-742, Korea
Email: jhhong00@snu.ac.kr

Ngaiming Mok
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email: nmok@hkucc.hku.hk

DOI: https://doi.org/10.1090/S1056-3911-2012-00611-0
Received by editor(s): August 6, 2010
Received by editor(s) in revised form: July 31, 2011, January 7, 2011, February 3, 2011, and June 2, 2011
Published electronically: November 5, 2012
Additional Notes: The first author’s work was supported by Research Settlement Fund for the new faculty of College of Natural Sciences in SNU 301-20070006. The second author’s research was partially supported by GRF grant HK7039/06P of the Research Grants Council of Hong Kong.

American Mathematical Society