Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

Schubert varieties are irreducible subvarieties of homogeneous manifold, which are important to understand the geometry of homogeneous manifold G/P and the action of the semisimple Lie group G. Consider the space of effective cycles in G/P with homology class equal to an integral multiple of the homology class of a Schubert variety X_w of type w. We will show that for a smooth Schubert variety X_w in Hermitian symmetric space with trivial exceptions, this cycle space is simple: any irreducible subvariety X with homology class [X]=r[X_w] is again a Schubert variety of type w.