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Given a rational homogeneous manifold $S=G/P$ of Picard number one and aSchubert variety $S_0 $ of $S$, the pair $(S,S_0)$ is said to be homologicallyrigid if any subvariety of $S$ having the same homology class as $S_0$ must bea translate of $S_0$ by the automorphism group of $S$. The pair $(S,S_0)$ issaid to be Schur rigid if any subvariety of $ S$ with homology class equal to amultiple 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-homogeneouscases. 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.

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