## Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

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- by Jaehyun Hong PDF
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**359**(2007), 2361-2381 Request permission

## 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 $G$.## References

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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
- Received by editor(s): October 26, 2004
- Received by editor(s) in revised form: April 13, 2005
- Published electronically: June 13, 2006
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2276624