2024/08/17 by Oliver Goertsches, Goertsches, Oliver, Eugenia Loiudice +1
Mathematics · #Algebraic and Geometric Analysis #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2408.09299
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 GKM3 setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM3 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 GKM3 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 ℍ Pn or the Grassmannian Gr2(ℂn) of complex 2-planes in ℂn.