Actions of Tori on Orbifolds

Haefliger, André; Salem, Eliane

doi:10.1007/bf02411354

Cited by 36 publications

(52 citation statements)

References 6 publications

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“…In this section we summarize the description of compact orbifolds with torus actions due to Orlik-Raymond [22] and Haefliger-Salem [13] (see also [14,18]). …”

Section: Torus Actions On Orbifoldsmentioning

confidence: 99%

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Toric self-dual Einstein metrics on compact orbifolds

Calderbank

Singer²

2006

Duke Math. J.

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“…In this section we summarize the description of compact orbifolds with torus actions due to Orlik-Raymond [22] and Haefliger-Salem [13] (see also [14,18]). …”

Section: Torus Actions On Orbifoldsmentioning

confidence: 99%

“…As we shall explain in section 1, compact simply connected 4-orbifolds X with an effective action of a 2-torus G = T 2 may be classified by work of Orlik-Raymond [22] and HaefligerSalem [13]. The orbit space W = X/G is a polygonal disc whose edges C 0 , C 1 , .…”

Section: Introductionmentioning

confidence: 99%

Toric self-dual Einstein metrics on compact orbifolds

Calderbank

Singer²

2006

Duke Math. J.

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“…Boundary orbits and fixed points. We now describe the orbifold charts about a boundary orbit and then about a fixed point following [11]. Let x ∈ M be contained in a boundary orbit, let…”

Section: Tensors On Orbifoldsmentioning

confidence: 99%

“…Simply connected orbifolds. It follows from the work of Haefliger and Salem in [11] that M is simply connected (π orb 1 (M ) = 0) when: (i) the orbit space N is topologically a closed disc; (ii) there are no exceptional orbits; (iii) and the elements of Λ associated to the components of the boundary orbits generate Λ.…”

Section: Tensors On Orbifoldsmentioning

confidence: 99%

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Compact anti-self-dual orbifolds with torus actions

Wright

2010

Sel. Math. New Ser.

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Abstract. We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.

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Some Examples of Toric Sasaki—Einstein Manifolds

Coevering

2009

Riemannian Topology and Geometric Structures on Manifolds

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Abstract. A series of examples of toric Sasaki-Einstein 5-manifolds is constructed which first appeared in the author's Ph.D. thesis [40]. These are submanifolds of the toric 3-Sasakian 7-manifolds of C. Boyer and K. Galicki. And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [8,7] an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM∞, where M∞ ∼ = S 2 × S 3 , for k odd. We then make use of the same framework to construct positive Ricci curvature toric Sasakian metrics on the manifolds X∞#kM∞ appearing in the classification of simply connected smooth 5-manifolds due to Smale and Barden. These manifolds are not spin, thus do not admit Sasaki-Einstein metrics. They are already known to admit toric Sasakian metrics (cf. [9]) which are not of positive Ricci curvature. We then make use of the join construction of C. Boyer and K. Galicki first appearing in [6], see also [9], to construct infinitely many toric Sasaki-Einstein manifolds with arbitrarily high second Betti number of every dimension 2m + 1 ≥ 5. This is in stark contrast to the analogous case of Fano manifolds in even dimensions. IntroductionA new series of quasi-regular Sasaki-Einstein 5-manifolds is constructed. These examples first appeared in the author's Ph.D. thesis [40]. They are toric, and arise as submanifolds of toric 3-Sasakian 7-manifolds. Applying 3-Sasakian reduction to torus actions on spheres C. Boyer, K. Galicki, et al [8] produced infinitely many toric 3-Sasakian 7-manifolds. More precisely, one has a 3-Sasakian 7-manifold S Ω for each integral weight matrix Ω satisfying some conditions to ensure smoothness. This produces infinitely many examples of each b 2 (S Ω ) ≥ 1. A result of D. Calderbank and M. Singer [12] shows that, up to finite coverings, this produces all examples of toric 3-Sasakian 7-manifolds. Associated to each S Ω is its twistor space Z, a complex contact Fano 3-fold with orbifold singularities. The action of T 2 complexifies to T 2 C = C * × C * acting on Z. Furthermore, if L is the line bundle of the complex contact structure, the action defines a pencilwhere t C is the Lie algebra of T 2 C . We determine the structure of the divisors in E. The generic X ∈ E is a toric variety with orbifold structure whose orbifold anti-canonical bundle K −1 X is positive. The total space M of the associated S 1 orbifold bundle to K X has a natural Sasaki-Einstein structure. Associated to any [3]) that M is diffeomorphic to #k(S 2 × S 3 ) where b 2 (M ) = k. We have the following: Theorem 1.1. Associated to every simply connected toric 3-Sasakian 7-manifold S is a toric quasi-regular Sasaki-. This gives an invertible correspondence. That is, given either X or M in diagram 1.1 one can recover the other spaces with their respective geometries.This gives an infinite series of quasi-regular Sasaki-Einstein structures on #m(S 2 × S 3 ) for every odd m ≥ 3.In section...

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Actions of Tori on Orbifolds

Cited by 36 publications

References 6 publications

Toric self-dual Einstein metrics on compact orbifolds

Toric self-dual Einstein metrics on compact orbifolds

Compact anti-self-dual orbifolds with torus actions

Some Examples of Toric Sasaki—Einstein Manifolds

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