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.

References in corpus (1)

• Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root