Orbit spaces of maximal torus actions on oriented Grassmannians of planes

TL;DR

This paper describes the orbit space and moment map of oriented Grassmannians of planes under maximal torus actions, using algebraic geometry tools to extend previous work on complex Grassmannians.

Contribution

It provides a new geometric and algebraic description of the orbit spaces of oriented Grassmannians of planes, building on and generalizing prior results.

Findings

Orbit spaces characterized via moment maps

Realization of Grassmannians as complex quadric hypersurfaces

Application of Geometric Invariant Theory to these actions

Abstract

Motivated by Buchstaber's and Terzic' work on the complex Grassmannians G(2,4) and G(2,5) we describe the moment map and the orbit space of oriented Grassmannians of planes under the action of a maximal compact torus. Our main tool is the realisation of these oriented Grassmannians as smooth complex quadric hypersurfaces and the relatively simple Geometric Invariant Theory of the corresponding algebraic torus action.