Orbit spaces for torus actions on Hessenberg varieties

In this paper we study effective actions of the compact torus Tn-1 on smooth compact manifolds M-2n of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to Sn+1 \(U-1 (sic)center dot center dot center dot(sic)U-l), the complement to the union of disjoint open subsets of the (n + 1)-sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type.

Automated summary

This paper investigates the effective actions of the compact torus Tn-1 on smooth compact manifolds M-2n of even dimension with isolated fixed points. It is shown that under certain conditions, the orbit space of such an action forms a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to a certain sphere, with applications to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices.