Bisymplectic Grassmannians of planes
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Bisymplectic Grassmannians of planes Vladimiro Benedetti1 Received: 27 September 2018 / Accepted: 29 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The bisymplectic Grassmannian I2 Gr(k, V ) parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when k = 2; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of I2 Gr(2, C6 ), which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology.
1 Introduction In complex algebraic geometry, classical Grassmannians are a special kind of homogeneous spaces for classical groups. They have been studied thoroughly for more than a century from different point of views: their geometry is governed by a rich combinatorial description, which manifests itself in many classical results about their cohomology. Moreover, the homogeneity condition has been very useful to investigate further properties of these varieties, such as their equivariant and quantum cohomology (see for instance [4,7,11,17]). Among classical Grassmannians, symplectic (respectively orthogonal) ones parametrize subspaces of a given vector space which are isotropic with respect to a non-degenerate symplectic (resp. orthogonal) form. Even for varieties which admit an action of a sufficiently big algebraic group, when the homogeneity hypothesis is dropped less is known: some efforts have led to the notion of GKM varieties (for the action of tori with a finite number of zero- and onedimensional orbits on complete varieties, they are defined in [8]) and some results have been obtained for specific examples (for instance, see [9,15,16]). In this paper, we present a work on a particular class of varieties, called bisymplectic Grassmannians, which are not homogeneous but admit an action of a big torus.
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Vladimiro Benedetti [email protected] Département de mathématiques et applications, ENS CNRS, PSL University, 75005 Paris, France
123
Journal of Algebraic Combinatorics
In general, one can define multisymplectic (respectively multiorthogonal) Grassmannians as the varieties parametrizing subspaces of a given vector space which are isotropic with respect to a fixed number of general symplectic (resp. orthogonal) forms. As an example, consider the unique Fano threefold of degree 22, which is usually denoted by V22 , and that appears in Iskovskikh’s classification (see [10]); Mukai showed that it can be seen as a trisymplectic Grassmannian I3 Gr(3, 7) of 3-dimensional subspaces of C7 . Of course, in general, asking the isotropy condition with respect to many symplectic forms implies that the corresponding Grassmannian is no longer homogeneous. How
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