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Rigidity of smooth Schubert varieties in Hermitian symmetric spaces


Author: Jaehyun Hong
Journal: Trans. Amer. Math. Soc. 359 (2007), 2361-2381
MSC (2000): Primary 14C25, 32M15, 14M15
DOI: https://doi.org/10.1090/S0002-9947-06-04041-4
Published electronically: June 13, 2006
MathSciNet review: 2276624
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Abstract: In this paper we study the space $ \mathcal{Z}_k(G/P, r[X_w])$ of effective $ k$-cycles $ X$ in $ G/P$ with the homology class equal to an integral multiple of the homology class of Schubert variety $ X_w$ of type $ w$. When $ X_w$ is a proper linear subspace $ \mathbb{P}^k$ $ (k<n)$ of a linear space $ \mathbb{P}^n$ in $ G/P \subset \mathbb{P}(V)$, we know that $ \mathcal{Z}_k(\mathbb{P}^n, r[\mathbb{P}^k])$ is already complicated. We will show that for a smooth Schubert variety $ X_w$ in a Hermitian symmetric space, any irreducible subvariety $ X$ with the homology class $ [X]=r[X_w]$, $ r\in \mathbb{Z}$, is again a Schubert variety of type $ w$, unless $ X_w$ is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space $ G/P$ is obtained by the action of the Lie group $ G$.


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Additional Information

Jaehyun Hong
Affiliation: Research Institute of Mathematics, Seoul National University, San 56-1 Sinrim-dong Kwanak-gu, Seoul, 151-747 Korea
Email: jhhong@math.snu.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-06-04041-4
Keywords: Analytic cycles, Hermitian symmetric spaces, Schubert varieties
Received by editor(s): October 26, 2004
Received by editor(s) in revised form: April 13, 2005
Published electronically: June 13, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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