A Morse-Bott unification of the Grassmannians of a symplectic vector space

TL;DR

This paper introduces a Morse-Bott function on the Grassmannian of a symplectic vector space, revealing its critical points and stable manifolds, and unifying various Grassmannian types through symplectic geometry.

Contribution

It constructs a quadratic Morse-Bott function that unifies different Grassmannians in symplectic geometry and describes their topology via gradient flow.

Findings

Critical loci are linear subspaces with isotropic and complex parts.

Stable manifolds match orbits of the linear symplectomorphism group.

Flow deformation retracts spaces onto compact homogeneous spaces.

Abstract

We construct a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. We show that its critical loci consist of linear subspaces that split into isotropic and complex parts and that its stable manifolds coincide with the orbits of the linear symplectomorphism group. These orbits generalize the Lagrangian, symplectic, isotropic, and coisotropic Grassmannians to include the Grassmannians of linear subspaces that are neither isotropic, coisotropic, nor symplectic. The negative gradient flow deformation retracts these spaces onto compact homogeneous spaces for the unitary group.