Structure theorem for compact Vaisman manifolds

Орнеа, Ливиу; Verbitsky, Misha

doi:10.4310/mrl.2003.v10.n6.a7

Cited by 60 publications

(71 citation statements)

References 7 publications

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“…If C(W min ) Z → M is the covering given by the minimal presentation of a compact Vaisman manifold and π the projection of the cone onto its radius, it is easy to check that π is equivariant with respect to the action of the covering maps on C(W min ) and the action of n ∈ Z on t ∈ R given by n + t. This describes in an alternative way the projection over S 1 of the Structure Theorem in [OV03], and moreover provides a structure theorem for other types of locally conformal structures, where the compactness of the base of the cone is given by other ways (see [IPP05] for an application to G 2 , Spin(7) and Spin(9) structures).…”

Section: Theorem 55 the Action Of A Connected Group G On A Compact mentioning

confidence: 99%

“…The main tool to prove the equivalence is the following Theorem, together with the Structure Theorem in [OV03].…”

Section: Vaisman Automorphisms and Reductionmentioning

confidence: 99%

“…This was implicit in [KO05] and was made explicit in [GOP05]. On the other hand, the Structure Theorem in [OV03] proves that any compact Vaisman manifold is a Riemannian suspension over a circle, with fibre a compact Sasakian manifold.…”

Section: Introductionmentioning

confidence: 99%

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Reduction of Vaisman structures in complex and quaternionic geometry

Gini¹,

Parton

Piccinni

2006

Journal of Geometry and Physics

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Abstract. We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair (K, Γ) where K is a Kähler manifold and Γ a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold (K, Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkähler reduction as an equivariant version of hyperkähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkähler reduction induces hyperkähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperkähler reduction.

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Section: Theorem 55 the Action Of A Connected Group G On A Compact mentioning

confidence: 99%

“…The main tool to prove the equivalence is the following Theorem, together with the Structure Theorem in [OV03].…”

Section: Vaisman Automorphisms and Reductionmentioning

confidence: 99%

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Reduction of Vaisman structures in complex and quaternionic geometry

Gini¹,

Parton

Piccinni

2006

Journal of Geometry and Physics

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“…Remark 5.2. The above result is an extension of [OV2], where the triviality of the adapted cohomology is derived directly from the structure theorem of compact Vaisman manifold [OV1]. For the general case of Riemannian foliations a similar attempt is not a trivial extension, as structural aspects of Riemannian foliations should be also considered [Mo].…”

Section: Corollary 54 For Any Vaisman Foliationmentioning

confidence: 99%

Basic Morse–Novikov cohomology for foliations

Slesar

2016

Math. Z.

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Abstract. In this paper we find sufficient conditions for the vanishing of the Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian manifolds. Several examples are presented, along with applications in the context of l.c.s. and l.c.K. foliations.

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“…Note that Boyer and Galicki discussed a similar problem in Section 8.2 of [8]. The product of S 1 with a Sasakian manifold admits geometric structures called a Vaisman manifold or a locally conformal Kähler manifold (see Ornea and Verbitsky [37]). Belgun [5] showed that both of structures are not stable under small deformations of complex structures.…”

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confidence: 99%

Deformation of Sasakian metrics

Nozawa

2013

Trans. Amer. Math. Soc.

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Abstract. Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics. We show that the triviality of the (0, 2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

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Structure theorem for compact Vaisman manifolds

Cited by 60 publications

References 7 publications

Reduction of Vaisman structures in complex and quaternionic geometry

Reduction of Vaisman structures in complex and quaternionic geometry

Basic Morse–Novikov cohomology for foliations

Deformation of Sasakian metrics

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