Upper bounds for the dimension of tori acting on GKM manifolds

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 𝒜(Γ, 𝛼, ∇), from an (abstract) (𝑚, 𝑛)-type GKM graph (Γ, 𝛼, ∇). 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 𝒜(Γ, 𝛼, ∇) has rank ℓ(> 𝑛) if and only if there exists an (𝑚, ℓ)-type GKM graph (Γ, \widetilde{𝛼}, ∇) which is an extension of (Γ, 𝛼, ∇). Using this combinatorial necessary and sufficient condition, we prove that the rank of 𝒜(Γ_𝑀, 𝛼_𝑀, ∇_𝑀) for the GKM graph (Γ_𝑀, 𝛼_𝑀, ∇_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 𝒜(Γ, 𝛼, ∇) of the complex Grassmannian of 2-planes 𝐺₂(ℂ^𝑛+2) with the natural effective 𝑇^𝑛+1-action, and prove that this action on 𝐺₂(ℂ^𝑛+2) is the maximal effective torus action which preserves the standard complex structure.