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April, 2019 Upper bounds for the dimension of tori acting on GKM manifolds
Shintarô KUROKI
J. Math. Soc. Japan 71(2): 483-513 (April, 2019). DOI: 10.2969/jmsj/79177917


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.

Funding Statement

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


Download Citation

Shintarô KUROKI. "Upper bounds for the dimension of tori acting on GKM manifolds." J. Math. Soc. Japan 71 (2) 483 - 513, April, 2019.


Received: 31 October 2017; Published: April, 2019
First available in Project Euclid: 1 March 2019

zbMATH: 07090053
MathSciNet: MR3943448
Digital Object Identifier: 10.2969/jmsj/79177917

Primary: 57S25
Secondary: 94C15

Keywords: GKM graph , GKM manifold , torus degree of symmetry

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 2 • April, 2019
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