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Rigidity of smooth Schubert varieties in Hermitian symmetric spaces
Authors:Jaehyun Hong
Affiliation:Research Institute of Mathematics, Seoul National University, San 56-1 Sinrim-dong Kwanak-gu, Seoul, 151-747 Korea
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]$, $ rin 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$.

Keywords:Analytic cycles   Hermitian symmetric spaces   Schubert varieties
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