Orbit spaces of torus actions on Hessenberg varieties

Abstract

We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to $S^{n+1}\setminus (U_1\bigsqcup \dots \bigsqcup U_l)$ where $S^{n+1}$ is the $(n+1)$--sphere and $U_1,\dots ,U_l$ are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.

arXiv comment15 pages

References in corpus (5)

• Topology of complexity one quotients
• Hessenberg varieties and hyperplane arrangements
• Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $C{P}^5$
• Geometry of regular Hessenberg varieties
• Hessenberg varieties, Slodowy slices, and integrable systems

Cited by in corpus (2)

• Torus actions of complexity one in non-general position
• Toric topology of the Grassmannian of planes in ${\mathbb{C}}^5$ and the del Pezzo surface of degree $5$