Orbit spaces of torus actions on Hessenberg varieties

2019/05/06 by Vladislav Cherepanov, Cherepanov, V.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1905.02294

openalex publication_date 2019/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider effective actions of a compact torus T^n-1 on an even-dimensional smooth manifold M²ⁿ 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 Sⁿ⁺¹ ∖ (U₁ \sqcup … \sqcup U_l) where Sⁿ⁺¹ is the (n+1)--sphere and U₁, …, U_l are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.

Citations

• Topology and geometry of the canonical action of T⁴ on the complex Grassmannian G_4,2 and the complex projective space CP⁵
• Geometry of regular Hessenberg varieties

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• Orbit spaces of equivariantly formal torus actions of complexity one
• Localization and Specialization for Hamiltonian Torus Actions
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