Rigidity of Schubert classes in orthogonal Grassmannians
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 quadratic type) if every (resp. none) flag element in his definition is isotropic. Under the intersection pairing, the dual of a class of Grassmannian type is a class of quadratic type and vice versa. The author characterizes rigid and multi rigid Schubert class of Grassmannian and quadratic type. He also characterize all the rigid classes in \(OG(2,n)\) if \(n>8\). The study of rigidity is strongly motivated by the classical problem of determing whether a cohomology class can be represented by a smooth subvariety, respectively by a (positive) linear combination of classes of smooth subvarieties. Rigid (respectively multi rigid) singular Schubert varieties cannot be represented by a smooth subvariety (respectively by a positive linear combination of classes of smooth subvarieties). For a large set of non rigid Schubert classes, the author provides explicit smooth deformations of Schubert varieties using combinatorially defined varieties called restriction varieties.
scientific article; zbMATH DE number 6308416
scientific article