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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 \(G_2({\mathbb{C}}^{n+2})\).

MSC:

57S25 Groups acting on specific manifolds
94C15 Applications of graph theory to circuits and networks
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References:

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