Upper bounds for the dimension of tori acting on GKM manifolds

摘要

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,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 G_2(C~(n+2)) with the natural effective T~(n+1)-action, and prove that this action on G_2(C~(n+2)) is the maximal effective torus action which preserves the standard complex structure.