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Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space 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 P k (k < n) of a linear space P n in G/P ⊂ P(V), we know that Z k (P n , r[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 ∈ 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.