Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

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 $G$.