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

2020/05/04 by Hong, Jaehyun, Mok, Ngaiming
#14M15 #32G10 #53C30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.01366

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.

Related

• Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces
• Schur flexibility of cominuscule Schubert varieties
• Rigidity of smooth Schubert varieties in a rational homogeneous manifold\n associated to a short root
• Rigidity of smooth Schubert varieties in Hermitian symmetric spaces
• Multi-rigidity of Schubert classes in partial flag varieties