Journal of the Mathematical Society of Japan

Upper bounds for the dimension of tori acting on GKM manifolds

Shintarô KUROKI

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action which preserves the almost complex structure, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this combinatorial necessary and sufficient condition, we prove 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 which can act on $M^{2m}$ effectively. As one of the applications of this result, we compute the rank associated to $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of 2-planes $G_{2}(\mathbb{C}^{n+2})$ with the natural effective $T^{n+1}$-action, and prove that this action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action which preserves the standard complex structure.

Note

This work was supported by JSPS KAKENHI Grant Number 15K17531, 24224002, 17K14196.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 2 (2019), 483-513.

Dates
Received: 31 October 2017
First available in Project Euclid: 1 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1551430829

Digital Object Identifier
doi:10.2969/jmsj/79177917

Mathematical Reviews number (MathSciNet)
MR3943448

Zentralblatt MATH identifier
07090053

Subjects
Primary: 57S25: Groups acting on specific manifolds
Secondary: 94C15: Applications of graph theory [See also 05Cxx, 68R10]

Keywords
GKM graph GKM manifold torus degree of symmetry

Citation

KUROKI, Shintarô. Upper bounds for the dimension of tori acting on GKM manifolds. J. Math. Soc. Japan 71 (2019), no. 2, 483--513. doi:10.2969/jmsj/79177917. https://projecteuclid.org/euclid.jmsj/1551430829


Export citation

References

  • [1] E. Bolker, V. Guillemin and T. Holm, How is a graph like a manifold?, math.CO/0206103.
  • [2] T. Braden and R. MacPherson, From moment graphs to intersection cohomology, Math. Ann., 321 (2001), 533–551.
  • [3] 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.
  • [4] C. Escher and C. Searle, Torus actions, maximality and non-negative curvature, arXiv:1506.08685.
  • [5] P. Fiebig, Moment graphs in representation theory and geometry, “Schubert calculus (Osaka 2012)” Adv. Studies in Pure Math., 71 (2016), 75–96.
  • [6] 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.
  • [7] O. Goertsches and M. Wiemeler, Positively curved GKM-manifolds, Int. Math. Res. Notices. (2015), 12015–12041.
  • [8] M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math., 131 (1998), 25–83.
  • [9] 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.
  • [10] V. Guillemin, S. Sabatini and C. Zara, Balanced fiber bundles and GKM theory, Int. Math. Res. Not. IMNR, (2013), no. 17, 3886–3910.
  • [11] V. Guillemin and C. Zara, One-skeleta, Betti numbers, and equivariant cohomology, Duke Math. J., 107 (2001), 283–349.
  • [12] W. Y. Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975.
  • [13] S. Kuroki, Characterization of homogeneous torus manifolds, Osaka J. Math., 47 (2010), 285–299.
  • [14] S. Kuroki, Classification of torus manifolds with codimension one extended actions, Transform. Groups, 16 (2011), 481–536.
  • [15] 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.
  • [16] S. Kuroki and M. Masuda, Root systems and symmetries of torus manifolds, Transform. Groups, 22 (2017), 453–474.
  • [17] H. Maeda, M. Masuda and T. Panov, Torus graphs and simplicial posets, Adv. Math., 212 (2007), 458–483.
  • [18] J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, N.J., 1974.
  • [19] S. Takuma, Extendability of symplectic torus actions with isolated fixed points, RIMS Kokyuroku, 1393 (2004), 72–78.
  • [20] J. S. Tymoczko, Permutation actions on equivariant cohomology of flag varieties, Toric topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 365–384.
  • [21] T. Watabe, On the torus degree of symmetry of $SU(3)$ and $G_2$, Sci. Rep. Niigata Univ. Ser. A, 15 (1978), 43–50.
  • [22] M. Wiemeler, Torus manifolds with non-abelian symmetries, Trans. Amer. Math. Soc., 364 (2012), 1427–1487.
  • [23] B. Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math., 191 (2003), 259–297.