GKM actions on almost quaternionic manifolds
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917751976624128 |
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| author | Goertsches, Oliver Loiudice, Eugenia |
| author_facet | Goertsches, Oliver Loiudice, Eugenia |
| contents | 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 ${\mathbb{C}}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09299 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | GKM actions on almost quaternionic manifolds Goertsches, Oliver Loiudice, Eugenia Differential Geometry Algebraic Topology 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 ${\mathbb{C}}^n$. |
| title | GKM actions on almost quaternionic manifolds |
| topic | Differential Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2408.09299 |