## Journal of the Mathematical Society of Japan

### Upper bounds for the dimension of tori acting on GKM manifolds

Shintarô KUROKI

#### 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
First available in Project Euclid: 1 March 2019

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

#### 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

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