Manifolds of isospectral matrices and Hessenberg varieties

TL;DR

This paper investigates the geometric and topological properties of manifolds of Hermitian matrices with staircase form and simple spectrum, revealing their smooth structure, cohomology, and connections to Hessenberg varieties.

Contribution

It introduces a new class of manifolds $X_h$, proves their smoothness and cohomological properties, and establishes a link to Hessenberg varieties within algebraic geometry.

Findings

$X_h$ is a smooth, equivariantly formal manifold.

Odd degree cohomology of $X_h$ vanishes.

Connections between $X_h$ and Hessenberg varieties are established.

Abstract

We study the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology of $X_h$ are described using GKM-theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and the semisimple Hessenberg varieties well-known in algebraic geometry. Both the spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families which can be generalized to other submanifolds of the flag variety.