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TITLE Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one
KIAS AUTHORS Hong, Jaehyun
JOURNAL SELECTA MATHEMATICA-NEW SERIES, 2020
ARCHIVE arxiv:2005.01366
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 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-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 amultiple of the homology class of S-0 into a sumof 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 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 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 C*-action we prove a refined form of transversality to get a degeneration of Z into a sum of distinct translates of S-0.
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