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Article: Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one
Title | Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one |
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Authors | |
Keywords | Schubert varieties Schur rigidity C∗-actions Transverality |
Issue Date | 2020 |
Publisher | Birkhaeuser Verlag AG. The Journal's web site is located at http://link.springer.de/link/service/journals/00029/index.htm |
Citation | Selecta Mathematica, 2020, v. 26, p. article no. 41 How to Cite? |
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. 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 a multiple of the homology class of S0 into a sum of distinct translates of S0, and by observing that the arguments for the homological rigidity apply since any two translates of S0 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 T0 opposite to S0. By transversality of general translates, a general translate of Z intersects T0 transversely and the C∗-action associated with the stabilizer of T0 induces a degeneration of Z into a sum of translates of S0, 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 S0. |
Persistent Identifier | http://hdl.handle.net/10722/283729 |
ISSN | 2023 Impact Factor: 1.2 2023 SCImago Journal Rankings: 1.715 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Hong, J | - |
dc.contributor.author | Mok, N | - |
dc.date.accessioned | 2020-07-03T08:23:16Z | - |
dc.date.available | 2020-07-03T08:23:16Z | - |
dc.date.issued | 2020 | - |
dc.identifier.citation | Selecta Mathematica, 2020, v. 26, p. article no. 41 | - |
dc.identifier.issn | 1022-1824 | - |
dc.identifier.uri | http://hdl.handle.net/10722/283729 | - |
dc.description.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. 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 a multiple of the homology class of S0 into a sum of distinct translates of S0, and by observing that the arguments for the homological rigidity apply since any two translates of S0 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 T0 opposite to S0. By transversality of general translates, a general translate of Z intersects T0 transversely and the C∗-action associated with the stabilizer of T0 induces a degeneration of Z into a sum of translates of S0, 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 S0. | - |
dc.language | eng | - |
dc.publisher | Birkhaeuser Verlag AG. The Journal's web site is located at http://link.springer.de/link/service/journals/00029/index.htm | - |
dc.relation.ispartof | Selecta Mathematica | - |
dc.rights | This is a post-peer-review, pre-copyedit version of an article published in [insert journal title]. The final authenticated version is available online at: http://dx.doi.org/[insert DOI] | - |
dc.subject | Schubert varieties | - |
dc.subject | Schur rigidity | - |
dc.subject | C∗-actions | - |
dc.subject | Transverality | - |
dc.title | Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one | - |
dc.type | Article | - |
dc.identifier.email | Mok, N: nmok@hku.hk | - |
dc.identifier.authority | Mok, N=rp00763 | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1007/s00029-020-00571-9 | - |
dc.identifier.scopus | eid_2-s2.0-85086574765 | - |
dc.identifier.hkuros | 310658 | - |
dc.identifier.volume | 26 | - |
dc.identifier.spage | article no. 41 | - |
dc.identifier.epage | article no. 41 | - |
dc.identifier.isi | WOS:000540274600001 | - |
dc.publisher.place | Switzerland | - |
dc.identifier.issnl | 1022-1824 | - |