Which Hessenberg varieties are GKM? (Q6084383)

The authors work over the field of the complex numbers: by GKM theory -- from Goresky-Kottwitz-MacPherson -- they mean algebraic and combinatorial techniques used to compute the equivariant cohomology of suitable spaces with torus actions; a space is called GKM if such techniques can be applied to it. The goal of this paper is to analyze torus action on a particular class of varieties called \textit{Hessenberg varieties}, in order to determine which are GKM. Let \(G=\mathrm{GL}_n\). An Hessenberg variety \(\mathcal{H}(X,H)\) is a subvariety of the complete flag variety \(G/B\), parametrized by a \(n \times n\) linear operator \(X\) and a linear subspace \(H\) of the Lie algebra of \(G\), such that \(H\) contains \(\mathfrak{b} = \mathrm{Lie}B\) and that is it \(\mathfrak{b}\)-stable (by the bracket operation). The corresponding variety is defined as \[ \mathcal{H}(X,H) = \{ gB \in G/B \,\, \colon\,\, g^{-1}X g \in H\}. \] It is known that Hessenberg varieties are GKM when \(X\) is semisimple; however, when \(X\) is nilpotent such a variety does not have a ``big enough'' torus action to be GKM in general. Hence, the aim is to identify the largest subtorus acting on them. A few main results are as follows: first, it is shown that all \(\mathcal{H}(X,H)\) admit the action of at least a rank one subtorus. Then, a particular family of matrices \(X\) is identified -- called \textit{skeletal nilpotent} -- which always admit a \(\mathbf{C}^*\)-action with isolated fixed point. Another important statement is a description of a system of linear equations that are necessary to guarantee that, for an element \(gB \in \mathcal{H}(X,H)\), its entire orbit (under a particular subtorus) is still contained in the Hessenberg variety. Next, it is proven that many Hessenberg varieties are GKM: not only those admitting a full torus action, but also any subvariety of a GKM space that itself carries a full torus action. Finally, the authors establish families of Hessenberg varieties that are GKM with respect to subtori \(K \subset T\) but not \(T\)-stable, and thus cannot be unions of Schubert varieties.