Grassmannians of symplectic subspaces
Let \((V, \omega )\) be a symplectic vector space of dimension \(2n\). The set of all \(2k\)-dimensional linear subspaces in \(V\) which are symplectic with respect to \(\omega \) is called the symplectic Grassmannian \(Gr^{Sp}(2k, 2n)\). The objective of this paper is to study the geometry of \(Gr^{Sp}(2k, 2n)\). A main tool introduced by the authors is the symplectic twistor Grassmannian \(\hat{{\mathcal T}}(2k, 2n)\) which is the set of triples \((S, J_S, J_{S^{\omega }})\) with \(S\in Gr^{Sp}(2k, 2n)\) and \(J_S\) (respectively \(J_{S^{\omega }}\)) a compatible complex structure on \(S\) (respectively \(S^{\omega }\)). Symplectic and Lagrangian submanifolds are then obtained in \(Gr^{Sp}(2k, 2n)\) as special kinds of Schubert cycles.
scientific article; zbMATH DE number 5968966