Bisymplectic Grassmannians of planes

TL;DR

This paper investigates the equivariant cohomology of bisymplectic Grassmannians of planes, deriving a Chevalley formula and analyzing specific cases to understand their geometric and algebraic structure.

Contribution

It provides the first equivariant Chevalley formula for bisymplectic Grassmannians of planes and explores their deformations and cohomology presentations.

Findings

Derived an equivariant Chevalley formula for hyper-plane class multiplication.

Analyzed the case of I$_2$Gr$(2, \

studied deformations and cohomology of the specific quasi-homogeneous variety.

Abstract

The bisymplectic Grassmannian I$_2$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 variety which admits an action of a torus with a finite number of fixed points. In this work we study its equivariant cohomology when $k = 2$; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyper-plane class by any Schubert class. Moreover, we study in detail the case of I$_2$Gr$(2, \mathbb{C}^6)$, which is a quasi-homogeneous variety, we analyze its deformations and we give a presentation of its cohomology.