Rigidity of Schubert classes in orthogonal Grassmannians
type: Item
instance of: scholarly article
title: Rigidity of Schubert classes in orthogonal Grassmannians
zbMATH Open document ID: 1354.14073
author: Izzet Coskun
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: Let \(G\) be an algebraic group over the complex and let \(P\) be a parabolic subgroup. A Schubert class \(\sigma\) in cohomology of \(X=G/P\) is called rigid if the only projective subvarieties of \(X\) representing \(\sigma\) are Schubert varieties. A Schubert class \(\sigma\) is called multi rigid if the only projective subvarieties of \(X\) representing a positive integral multiple \(k\sigma\) are unions of \(k\) Schubert varieties. In recent years, there has been significant progress in determining multi rigid Schubert classes in compact complex Hermitian symmetric spaces by using differential geometric methods. In [J. Differ. Geom. 87, No. 3, 493--514 (2011; Zbl 1232.14032)] the author characterized the rigid Schubert classes in Grassmannians, using algebro-geometric techniques. In this paper he extends such techniques to orthogonal Grassmannians. Remark that these varieties are not in general compact complex Hermitian symmetric spaces. The algebro-geometric techniques have two main advantages over the differential geometric techniques. First, they can prove the rigidity of a class even when the class is not multi rigid. Second, they apply to homogeneous varieties other than compact complex Hermitian symmetric spaces. The main disadvantage is that studying multiples of Schubert classes gets successively more difficult as the multiple increases, making the technique less suitable for studying multi rigidity. In a Grassmannian \(G(k,n)\) the Schubert classes are the classes of the Schubert varieties, which are the closure of the locus constructed by fixing the dimensions of the intersection of \(V\in G(k,n)\) with the subspaces of a fixed flag. To obtain the Schubert varieties of the odd orthogonal Grassmanian \(OG(k,2n+1)\) one has to start with an isotropic flag. The case of even orthogonal is slightly more technical; moreover the author allow \(OG(k,2k)\) do have 2 connected components. A Schubert class in \(OG(k,n)\) is of Grassmannian type (resp. of quadrat…
reviewed by: Alessandro Ruzzi
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 6308416
zbMATH Keywords: Schubert class
zbMATH Keywords: orthogonal Grassmannian
MaRDI profile type: Publication
full work available at URL: s11856-014-0009-3
OpenAlex ID: W2061340086
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DOI: 10.1007/S11856-014-0009-3
description: scientific article; zbMATH DE number 6308416