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arXiv · 2005.01366

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

Jaehyun Hong·Ngaiming Mok
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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.

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Jaehyun Hong, Ngaiming Mok. 2020-05-04. Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one. https://arxiv.org/abs/2005.01366

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