Abstract. In this paper we study the space Zk(G/P, r[Xw]) of effective k-cycles X in G/P with the homology class equal to an integral multiple of the homology class of Schubert variety Xw of type w. When Xw is a proper linear subspace Pk (k<n) of a linear space Pn in G/P ⊂ P(V), we know that Zk(Pn,r[Pk]) is already complicated. We will show that for a smooth Schubert variety Xw in a Hermitian symmetric space, any irreducible subvariety X with the homology class [X] =r[Xw], r ∈ Z, is again a Schubert variety of type w, unless Xw 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. 1
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