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, α, ∇) which is an extension of (Γ, α, ∇). Using this combinatorial necessarily 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 applications of this result, we compute the rank of A(Γ, α, ∇) of the complex Grassmannian of 2-planes G 2 (C n+2 ) with some effective T n+1 -action, and prove that the T n+1 -action on G 2 (C n+2 ) is the maximal effective torus action which preserves the standard complex structure.Theorem 1.2. Let (Γ, α, ∇) be an abstract (m, n)-type GKM graph. Then, the following two statements are equivalent:(1) rk A(Γ, α, ∇) ≥ ℓ for some n ≤ ℓ ≤ m;(2) there is an (m, ℓ)-type GKM graph (Γ, α, ∇) which is an extension of (Γ, α, ∇).Because a GKM manifold (M 2m , T n ) defines an (m, n)-type GKM graph (see Section 4), Theorem 1.2 implies that the maximal dimension of torus which can act on a GKM manifold M is bounded from above by the rank of the group of axial functions of the GKM graph induced from M . Namely, we obtain the main result of this paper as follows (see Section 4 for details):Then, the T n -action on M 2m does not extend to any T ℓ+1 -action preserving the given almost complex structure.Hence, if rk A(Γ M , α M , ∇ M ) = n, then the T n -action on M 2m is maximal among torus actions which preserve the given almost complex structure.Remark 1.4. Shunji Takuma also obtains a partial answer to Problem 1.1 by introducing an obstruction class for the extension of an (m, n)-type GKM graph to an (m, n + 1)-type GKM graph in his note [19]. Theorem 1.2 may be regarded as the generalization of his result. Problem 1.1 is reminiscent of the computation of the torus degree of symmetry of a manifold X (see [12]), i.e., the maximal dimension of a torus which can act on X effectively. A torus degree of symmetry has been studied for many classes of manifolds, in particular from differential geometry (see e.g. [4,12,21,23]). Corollary 1.3 may be regarded as to give an upper bound of the torus degree of symmetry of an invariant almost complex structure of a GKM manifold. As an application of Corollary 1.3, in the final section (Section 5), we compute the torus degree of such symmetry for the complex Grassmannian of 2-planes, denoted as G 2 (C n+2 ) ≃ GL(n + 2, C)/GL(2, C) × GL(n, C) ≃ U (n + 2)/U (2) × U (n). Namely, we compute rk A(Γ M , α M , ∇ M ) for ...

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

“…Is this Section we apply the GKM-theory in order to study torus actions. Throughout this Section, we follow [24] and adopt the GKM-notions to the case of a torus action on an algebraic variety with isolated fixed points having 2-dependent weights. 6.1.…”

“…Definition 6.2 (cf. [24]). We call α an axial function, if Remind that if Γ is a graph and one requires pairwise linear independence of values of α(e), where e ∈ E v (Γ) for any vertex v ∈ V(Γ), then (Γ, α) is called a GKM-graph.…”

“…For example, the natural action of T n+1 : Gr 2 (C n+2 ) is maximal (in a sense that there is no equivariant embedding of T n+1 to the torus of greater dimension, acting on Gr 2 (C n+2 )), since the group Aut Gr 2 (C n+2 ) of automorphisms of the Grassmanian of 2-planes in C n+2 is algebraic and has rank n + 1 (see [11], cf. [24]).…”

“…The converse also holds ( [29]), however the binomiality property is not easy to verify. Secondly, there is no effective T k -action on X, provided that one knows that the group Aut X is algebraic and has rank less than k. Third method is the estimate on the dimension of a torus extending a given GKM-action, based upon the application of GKM-theory [24]. Finally, we mention the theory of Cox rings which reduces the study of automorphisms of algebraic varieties to the study of graded factorial algebras' automorphisms [1].…”

GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

“…After the works of Guillemin-Zara, a GKM graph can be regarded as a combinatorial approximation of a space with torus action, and it has been studied by some mathematicians, e.g. see [MMP,GHZ,GSZ,FIM,FY,Ku19,DKS]. In this paper, we introduce a certain class of GKM graphs with legs and attempt to unify two slightly different classes of manifolds from GKM theoretical point of view, i.e., toric hyperKähler manifolds and cotangent bundles of toric manifolds, where a leg is a half-line whose boundary corresponds to the initial vertex.…”