Upper bounds for the dimension of tori acting on GKM manifolds. (English) Zbl 1421.57047

Let \(M^{2m}\) be a \(2m\)-dimensional, compact, connected, almost complex manifold, and let \(T^n\) be the \(n\)-dimensional torus. Assume \(T^n\) acts effectively on \(M^{2m}\), and that the action preserves the almost complex structure of \(M^{2m}\). The pair \((M^{2m}, T^n)\) is called a GKM manifold, if the fixed point set of the action is nonempty and isolated, and if the closure of each connected component of \(1\)-dimensional orbits is equivariantly diffeomorphic to the \(2\)-dimensional sphere. In this paper, the author considers the question when a GKM manifold \((M^{2m}, T^n)\) can be extended to a GKM manifold \((M^{2m}, T^l)\), where \(T^n\subset T^l\) and \(n<l\leq m\).
The author gives a combinatorial method to determine an upper bound for \(l\). To do that he introduces a free abelian group with finite rank, \({\mathcal{A}}(\Gamma, \alpha,\nabla)\), from an \((m,n)\)-type GKM graph \((\gamma,\alpha,\nabla)\) defined in the paper. He shows that \({\mathcal{A}}(\Gamma, \alpha,\nabla)\) has rank \(l>n\) if and only if there is an \((m,l)\)-type GKM graph \((\Gamma, \tilde{\alpha}, \nabla)\) extending \((\Gamma, \alpha,\nabla)\). As a corollary, he obtains the main result of the paper, which says that the rank of \({\mathcal{A}}(\Gamma_M,\alpha_M, \nabla_M)\) for the GKM graph \((\Gamma_M,\alpha_M, \nabla_M)\) induced from \((M^{2m}, T^n)\) gives an upper bound for the dimension of a torus that can act effectively on \(M^{2m}\).
The author applies the main result to compute the rank of \({\mathcal{A}}(\Gamma, \alpha, \nabla)\) for the complex Grassmannian of \(2\)-planes \(G_2({\mathbb{C}}^{n+2})\) with the standard effective \(T^{n+1}\)-action. He also shows that this action is the maximal effective torus action preserving the almost complex structure of \(G_2({\mathbb{C}}^{n+2})\).


57S25 Groups acting on specific manifolds
94C15 Applications of graph theory to circuits and networks
Full Text: DOI arXiv Euclid


[1] T. Braden and R. MacPherson, From moment graphs to intersection cohomology, Math. Ann., 321 (2001), 533-551. · Zbl 1077.14522
[2] V. M. Buchstaber and S. Terzić, Equivariant complex structures on homogeneous spaces and their cobordism classes, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser., 2, 224, Amer. Math. Soc., Providence, RI, 2008, 27-57. · Zbl 1170.57026
[3] C. Escher and C. Searle, Torus actions, maximality and non-negative curvature, arXiv:1506.08685. · Zbl 1416.53034
[4] P. Fiebig, Moment graphs in representation theory and geometry, “Schubert calculus (Osaka 2012)” Adv. Studies in Pure Math., 71 (2016), 75-96. · Zbl 1407.17010
[5] Y. Fukukawa, H. Ishida and M. Masuda, The cohomology ring of the GKM graph of a flag manifold of classical type, Kyoto J. Math., 54 (2014), 653-677. · Zbl 1436.55012
[6] O. Goertsches and M. Wiemeler, Positively curved GKM-manifolds, Int. Math. Res. Notices. (2015), 12015-12041. · Zbl 1339.57042
[7] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math., 131 (1998), 25-83. · Zbl 0897.22009
[8] V. Guillemin, T. Holm and C. Zara, A GKM description of the equivariant cohomology ring of a homogeneous space, J. Algebraic Combin., 23 (2006), 21-41. · Zbl 1096.53050
[9] V. Guillemin, S. Sabatini and C. Zara, Balanced fiber bundles and GKM theory, Int. Math. Res. Not. IMNR, (2013), no. 17, 3886-3910. · Zbl 1329.55005
[10] V. Guillemin and C. Zara, One-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J., 107 (2001), 283-349. · Zbl 1020.57013
[11] W. Y. Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. · Zbl 0429.57011
[12] S. Kuroki, Characterization of homogeneous torus manifolds, Osaka J. Math., 47 (2010), 285-299. · Zbl 1238.57033
[13] S. Kuroki, Classification of torus manifolds with codimension one extended actions, Transform. Groups, 16 (2011), 481-536. · Zbl 1246.57082
[14] S. Kuroki, An Orlik-Raymond type classification of simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology, Pacific J. of Math., 280 (2016), 89-114. · Zbl 1334.57036
[15] S. Kuroki and M. Masuda, Root systems and symmetries of torus manifolds, Transform. Groups, 22 (2017), 453-474. · Zbl 1416.57014
[16] H. Maeda, M. Masuda and T. Panov, Torus graphs and simplicial posets, Adv. Math., 212 (2007), 458-483. · Zbl 1119.55004
[17] J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, N.J., 1974. · Zbl 0298.57008
[18] S. Takuma, Extendability of symplectic torus actions with isolated fixed points, RIMS Kokyuroku, 1393 (2004), 72-78.
[19] J. S. Tymoczko, Permutation actions on equivariant cohomology of flag varieties, Toric topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 365-384. · Zbl 1147.14024
[20] T. Watabe, On the torus degree of symmetry of \(SU(3)\) and \(G_2\), Sci. Rep. Niigata Univ. Ser. A, 15 (1978), 43-50. · Zbl 0395.57026
[21] M. Wiemeler, Torus manifolds with non-abelian symmetries, Trans. Amer. Math. Soc., 364 (2012), 1427-1487. · Zbl 1244.57064
[22] B. Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math., 191 (2003), 259-297. · Zbl 1062.53029
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