Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one

Abstract

Given a rational homogeneous manifold S=G/P of Picard number one and a Schubert variety S0 of S, the pair (S,S0) is said to be homologically rigid if any subvariety of S having the same homology class as S0 must be a translate of S0 by the automorphism group of S. The pair (S,S0) is said to be Schur rigid if any subvariety of S with homology class equal to a multiple of the homology class of S0 must be a sum of translates of S0. Earlier we completely determined homologically rigid pairs (S,S0) in case S0 is homogeneous and answered the same question in smooth non-homogeneous cases. In this article we consider Schur rigidity, proving that (S,S0) exhibits Schur rigidity whenever S0 is a non-linear smooth Schubert variety. Modulo a classification result of the first author’s, our proof proceeds by a reduction to homological rigidity by deforming a subvariety Z of S with homology class equal to a multiple of the homology class of S0 into a sum of distinct translates of S0, and by observing that the arguments for the homological rigidity apply since any two translates of S0 intersect in codimension at least two. Such a degeneration is achieved by means of the C∗-action associated with the stabilizer of the Schubert variety T0 opposite to S0. By transversality of general translates, a general translate of Z intersects T0 transversely and the C∗-action associated with the stabilizer of T0 induces a degeneration of Z into a sum of translates of S0, not necessarily distinct. After investigating the Bialynicki-Birular decomposition associated with the C∗-action we prove a refined form of transversality to get a degeneration of Z into a sum of distinct translates of S0.