GKM actions on almost quaternionic manifolds

Abstract

We introduce quaternionic structures on abstract GKM graphs, as the combinatorial counterpart of almost quaternionic structures left invariant by a torus action of GKM type. In the GKM$_3$ setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM$_3$ actions on positive quaternion-Kähler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract GKM$_3$ graph with quaternionic structure satisfying this restriction on the 2-faces of the GKM graph is that of a torus action on quaternionic projective space ${\mathbb{H}} P^n$ or the Grassmannian ${\mathrm{Gr}}_2({\mathbb{C}}^n)$ of complex 2-planes in ${\mathbb{C}}^n$.