Non-negatively curved GKM orbifolds

Goertsches, Oliver; Wiemeler, Michael

doi:10.1007/s00209-021-02853-0

Cited by 5 publications

(6 citation statements)

References 28 publications

(52 reference statements)

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“…The existence of a compatible connection on the GKM graph associated to a GKM manifold was shown in [36, p. 6] and [28,Proposition 2.3]. We thus obtain: Proposition 2.37.…”

Section: The Abstract Notion Of a Gkm Graph And Geometric Structuresmentioning

confidence: 60%

“…Equivalently, one may formulate it as follows: for any sign choices of α(∇ e (f )), α(e), and α(f ) there exists c ∈ Z and ε ∈ {±1} such that α(∇ e (f )) = εα(f ) + cα(e). The smoothness condition is related to the integrality of the linear combination; one may generalize GKM theory for actions on orbifolds (see [36,28]), where rational constants occur.…”

Section: The Abstract Notion Of a Gkm Graph And Geometric Structuresmentioning

confidence: 99%

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Low-dimensional GKM theory

Goertsches¹,

Konstantis²,

Zoller³

2022

Preprint

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“…The existence of a compatible connection on the GKM graph associated to a GKM manifold was shown in [36, p. 6] and [28,Proposition 2.3]. We thus obtain: Proposition 2.37.…”

Section: The Abstract Notion Of a Gkm Graph And Geometric Structuresmentioning

confidence: 60%

Section: The Abstract Notion Of a Gkm Graph And Geometric Structuresmentioning

confidence: 99%

Low-dimensional GKM theory

Goertsches¹,

Konstantis²,

Zoller³

2022

Preprint

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show abstract

“…Given an action of a torus on a connected, compact manifold satisfying the GKM conditions, the GKM graph of the action admits a compatible connection, see [GW22, Proposition 2.3] or [GZ01]. Note that there exist different conventions in the literature of whether the connection is part of the structure of an abstract GKM graph or not.…”

Section: Gkm Theory and Geometric Structuresmentioning

confidence: 99%

Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions

Goertsches,

Konstantis,

Zoller

2023

Compositio Math.

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We classify fibrations of abstract $3$ -regular GKM graphs over $2$ -regular ones, and show that all fibrations satisfying the known necessary conditions for realizability are, in fact, realized as the projectivization of equivariant complex rank- $2$ vector bundles over quasitoric $4$ -manifolds or $S^4$ . We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way, we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with a prescribed number of isolated fixed points.

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“…We remark that if 2 ≤ k ≤ n many elements in Definition 2.1 (3) are linearly independent and G is compact torus then X is called a GKM korbifold in [GW20]. Next we introduce the concept of simplicial orbifold complexes generalizing the definition of simplicial complexes.…”

Section: Simplicial Gkm Orbifold Complexes and Its Generalized Cohomo...mentioning

confidence: 99%

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

Koushik¹,

Sarkar²

2021

Preprint

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In this paper, we define 'simplicial GKM orbifold complexes' and 'simplicial GKM graph complexes. We study some of their topological and combinatorial properties. We discuss some necessary conditions to confirm the invariant q-cell structure on a simplicial GKM orbifold complex. We prove Thom isomrphism for general orbifold Gbundles for equivariant cohomology and equivariant K-theory with rational coefficients. We use this to extend the main result of Harada-Henriques-Holm (2005) to the category of G-spaces equipped with 'singular invariant stratification'. Then we introduce an equivalent definition of weighted Grassmann orbifolds as a generalization of Grassmann manifolds which were discussed by Corti and Reid (2002) and explicitly studied by Abe and Matsumura (2015) to compute their equivariant cohomology. We study their different q-cell structures and discuss some necessary condition to determine if they have invariant CW-structures. We employ the techniques from the first part to study some generalized cohomology theories of weighted Grassmann orbifolds. We introduce 'divisive' weighted Grassmann orbifolds and compute their equivariant cohomology, equivariant K-theory and equivariant cobordism ring with integer coefficients. We discuss Schubert calculus for the equivariant cohomology ring and show how to compute the corresponding structure constants with integral coefficients.

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Non-negatively curved GKM orbifolds

Cited by 5 publications

References 28 publications

Low-dimensional GKM theory

Low-dimensional GKM theory

Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions

Integral generalized equivariant cohomologies of weighted Grassmann orbifolds

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