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 $S_0$ of S, the pair $(S,S_0)$ is said to be homologically rigid if any subvariety of S having the same homology class as $S_0$ must be a translate of $S_0$ by the automorphism group of S. The pair $(S,S_0)$ is said to be Schur rigid if any subvariety of S with homology class equal to a multiple of the homology class of $S_0$ must be a sum of translates of $S_0$. Earlier we completely determined homologically rigid pairs $(S,S_0)$ in case $S_0$ is homogeneous and answered the same question in smooth non-homogeneous cases. In this article we consider Schur rigidity, proving that $(S,S_0)$ exhibits Schur rigidity whenever $S_0$ 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 $S_0$ into a sum of distinct translates of $S_0$, and by observing that the arguments for the homological rigidity apply since any two translates of $S_0$ intersect in codimension at least two. Such a degeneration is achieved by means of the ${\mathbb {C}}^*$-action associated with the stabilizer of the Schubert variety $T_0$ opposite to $S_0$. By transversality of general translates, a general translate of Z intersects $T_0$ transversely and the ${\mathbb {C}}^*$-action associated with the stabilizer of $T_0$ induces a degeneration of Z into a sum of translates of $S_0$, not necessarily distinct. After investigating the Bialynicki-Birular decomposition associated with the ${\mathbb {C}}^*$-action we prove a refined form of transversality to get a degeneration of Z into a sum of distinct translates of $S_0$.