TOPOLOGY AND GEOMETRY OF THE CANONICAL ACTION OF T-4 ON THE COMPLEX GRASSMANNIAN G(4,2) AND THE COMPLEX PROJECTIVE SPACE CP5

摘要

We consider the canonical action of the compact torus T-4 on the complex Grassmann manifold G(4,2) and prove that the orbit space G(4,2)/T-4 is homeomorphic to the sphere S-5. We prove that the induced map from G(4,2) to the sphere S-5 is not smooth and describe its smooth and singular points. We also consider the action of T-4 on CP5 induced by the composition of the second symmetric power representation of T-4 in T-6 and the standard action of T-6 on CP5 and prove that the orbit space CP5/T-4 is homeomorphic to the join CP2 * S-2. The Plucker embedding G(4,2) subset of CP5 is equivariant for these actions and induces the CP1*S-2 subset of CP2*S-2 embedding for the standard embedding CP1 subset of CP2.