Balanced Fiber Bundles and GKM Theory

Guillemin, Victor; Sabatini, Silvia; Zara, Catalin

doi:10.1093/imrn/rns168

Cited by 7 publications

(5 citation statements)

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“…Figure 4 shows how the single edge (drawn in boldface) is transported around the cycle with a Möbius-like twist on the back edges (drawn with dotted lines). This is part of a general phenomenon studied extensively by Guillemin-Sabatini-Zara [21,22]. Each Grassmannian G(k, n) can be identified with the quotient GL n (C)/P k of the flag variety by the larger group P k ⊃ B of n × n invertible matrices whose bottom-left n − k × n − k block is zero.…”

Section: Permutations At Verticesmentioning

confidence: 97%

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Splines in geometry and topology

Tymoczko

2016

Computer Aided Geometric Design

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Abstract. This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.The goal of this survey paper is to describe how splines arise in geometry and topology. Geometric splines usually appear under the name GKM theory after Goresky-Kottwitz-MacPherson, who developed them to compute what is called the cohomology ring of a geometric object. Geometers and analysts ask many of the same questions about splines: what is their dimension? can we identify a basis? can we find explicit formulas for the elements of the basis? However geometric constraints can change the tone of these questions: the splines may satisfy various symmetries or have a basis satisfying certain conditions. And some questions are specific to geometric splines: geometers particularly care about the multiplication table with respect to a given basis.In Section 1 we discuss GKM theory, followed in Section 2 by some important families of geometric examples, some of which are well-known to students of analytic splines and some of which may be useful in future. Section 3 sketches some techniques that are natural from the perspective of a geometer/topologist, including Morse flows and symmetries that come from geometric representation theory. Finally in Section 4 we generalize splines to a more abstract ring setting, both as a useful conceptual framework and because it provides new combinatorial tools. This paper is targeted at researchers in geometric design, especially those with an analytic background. Our aim is to give an overview of theoretical tools and techniques from geometry and topology; we often illustrate concepts by example and refer to the literature for details on technical aspects. A reader with a different mathematical perspective may be interested in surveys like [27,35,36]. GKM theoryCohomology is an algebraic gadget associated to a geometric object X that encodes various properties of X. Among other things, the cohomology of X indicates the dimension of X, the number of connected components (how many separate pieces X has), how many holes X has, whether X has singularities, and how different subspaces of X intersect.We could treat very general kinds of geometric objects but for simplicity in this survey we take X to be a compact complex manifold. When we say that cohomology is an "algebraic gadget" the most general interpretation is that cohomology is a ring. In the cases of interest here, the cohomology ring is actually an algebra, namely a vector space in which one can multiply vectors. The technical condition we assume is that cohomology has coefficients in Q, R, or especially C.In fact we will consider an enhanced version of cohomology called T -equivariant cohomology. Equivariant cohomology has strictly more information than ordinary cohomology yet surprisingly can be easier to compute. In our case T is a torus, ...

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Section: Permutations At Verticesmentioning

confidence: 97%

“…Figure 11 shows the symmetrized basis for splines on P 2 as a comparison. Guillemin-Sabatini-Zara constructed examples of symmetrized bases as a step towards identifying bases for splines that arise as fiber bundles [21,22].…”

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

Splines in geometry and topology

Tymoczko

2016

Computer Aided Geometric Design

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Kuroki¹,

Uma²

2021

Preprint

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We introduce a class of labeled graphs (with legs) which contains two classes of GKM graphs of 4n-dimensional manifolds with T n ×S 1 -actions, i.e., GKM graphs of the toric hyperKähler manifolds and of the cotangent bundles of toric manifolds. Under some conditions, the graph equivariant cohomology ring of such a labeled graph is computed. We also give a module basis of the graph equivariant cohomology by using a shelling structure of such a labeled graph, and study their multiplicative structure.

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“…This leads us to the study of geometric and topological properties of GKM manifolds using combinatorial properties of GKM graphs (see e.g. [6,7,9,10,15,17,19,20] etc). In this paper, we introduce a new invariant of GKM graphs and provide a partial answer to the extension problem of torus actions on GKM manifolds.…”

Section: Introductionmentioning

confidence: 99%

Upper bounds for the dimension of tori acting on GKM manifolds

Kuroki¹

2019

J. Math. Soc. Japan

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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 ...

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Balanced Fiber Bundles and GKM Theory

Cited by 7 publications

References 7 publications

Splines in geometry and topology

Splines in geometry and topology

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

Upper bounds for the dimension of tori acting on GKM manifolds

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