GKM theory for torus actions with non-isolated fixed points

TL;DR

This paper extends GKM theory to torus actions with non-isolated fixed points, showing that the fixed point components are diffeomorphic and that equivariant cohomology can be computed from associated combinatorial data.

Contribution

It introduces a new class of GKM actions with non-isolated fixed points and establishes analogous cohomological computation methods.

Findings

All fixed point components are diffeomorphic.

Equivariant cohomology can be derived from a combinatorial graph.

The framework generalizes classical GKM theory to broader fixed point sets.

Abstract

Let $M^{2d}$ be a compact symplectic manifold and $T$ a compact $n$-dimensional torus. A Hamiltonian action, $\tau$, of $T$ on $M$ is a GKM action if, for every $p \in M^T$, the isotropy representation of $T$ on $T_pM$ has pair-wise linearly independent weights. For such an action the projection of the set of zero and one-dimensional orbits onto $M/T$ is a regular $d$-valent graph; and Goresky, Kottwitz and MacPherson have proved that the equivariant cohomology of $M$ can be computed from the combinatorics of this graph. (See \cite{GKM:eqcohom}.) In this paper we define a ``GKM action with non-isolated fixed points'' to be an action, $\tau$, of $T$ on $M$ with the property that for every connected component, $F$ of $M^T$ and $ p \in F$ the isotropy representation of $T$ on the normal space to $F$ at $p$ has pair-wise linearly independent weights. For such an action, we show that all components of $M^T$ are diffeomorphic and prove an analogue of the theorem above.