A non-Sasakian Lefschetz $K$-contact manifold of Tievsky type

Cappelletti–Montano, Beniamino; Nicola, Antonio De; Marrero, Juan Carlos; Yudin, Ivan

doi:10.1090/proc/13187

Cited by 5 publications

(8 citation statements)

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“…Theorem 2. 5. Let (M, F, ω) be a transversely symplectic foliation with the transverse hard Lefschetz property.…”

Section: Hodge Theory On Transversely Symplectic Foliationsmentioning

confidence: 99%

“…). It is also noteworthy that there exist examples of compact K-contact manifolds which do not admit any Sasakian structures, and which satisfy the hard Lefschetz property as introduced in[4,5].By[21, Theorem 4.4], these non-Sasakian K-contact manifolds also satisfy the transverse hard Lefschetz property.Example 5.3 (Hamiltonian actions on contact manifolds). Let M be a 2n + 1 dimensional compact contact manifold with a contact one form η and a Reeb vector field ξ, and let G be…”

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confidence: 99%

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Equivariant formality of Hamiltonian transversely symplectic foliations

Lin

Yang

2019

Pacific J. Math.

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Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic dδ-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.

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“…Theorem 2. 5. Let (M, F, ω) be a transversely symplectic foliation with the transverse hard Lefschetz property.…”

Section: Hodge Theory On Transversely Symplectic Foliationsmentioning

confidence: 99%

mentioning

confidence: 99%

Equivariant formality of Hamiltonian transversely symplectic foliations

Lin

Yang

2019

Pacific J. Math.

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“…On the other hand, in [5], we present an example of a compact Lefschetz contact manifold N which does not admit any Sasakian structure. So, the standard l.c.s.…”

Section: (Basic Lefschetz ⇒ Lefschetz)mentioning

confidence: 99%

“…Note that (1) follows using that b k (N ) is even if k is odd and k ≤ n, with dim N = 2n + 1 and b k (N ) the k-th Betti number of N . On the other hand, in [5], we present an example of a compact Lefschetz contact manifold N which does not admit any Sasakian structure. So, the standard l.c.s.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Hard Lefschetz theorem for Vaisman manifolds

Cappelletti–Montano

Nicola

Marrero

et al. 2018

Trans. Amer. Math. Soc.

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We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual 1-form coincides with the Lee 1-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics. 2010 Mathematics Subject Classification. Primary 53C25, 53C55, 53D35. (J.C.M.) and by Prin 2010/11 -Varietà reali e complesse: geometria, topologia e analisi armonica -Italy (B.C.M.), and by the exploratory research project in the frame of Programa Investigador FCT IF/00016/2013 (I.Y.). J.C.M. acknowledges the Centre for Mathematics of the University of Coimbra in Portugal for its support and hospitality in a visit where a part of this work was done.

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“…Using this it is possible to construct (non-simply connected) compact manifolds which are K-contact but not Sasakian [11]. Also it has been used to provide an example of a solvmanifold of dimension 5 which satisfies the hard Lefschetz property and which is K-contact and not Sasakian [9].…”

Section: Introductionmentioning

confidence: 99%

Homology Smale–Barden Manifolds with K-contact but not Sasakian Structures

Muñoz

Rojo

Tralle

2018

International Mathematics Research Notices

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Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with H 1 (M, Z) = 0 which is K-contact but which carries no semi-regular Sasakian structures.2010 Mathematics Subject Classification. 53C25, 53D35, 57R17, 14J25.

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A non-Sasakian Lefschetz $K$-contact manifold of Tievsky type

Abstract: Abstract. We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of K-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

Cited by 5 publications

References 20 publications

Equivariant formality of Hamiltonian transversely symplectic foliations

Equivariant formality of Hamiltonian transversely symplectic foliations

Hard Lefschetz theorem for Vaisman manifolds

Homology Smale–Barden Manifolds with K-contact but not Sasakian Structures

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