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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

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.

Funding Statement

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

Citation

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

Information

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

Subjects:
Primary: 57S25
Secondary: 94C15

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

Rights: Copyright © 2019 Mathematical Society of Japan

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