Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

Andrada, Adrián; Origlia, Marcos

doi:10.1007/s00229-017-0938-3

Cited by 34 publications

(71 citation statements)

References 36 publications

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“…Since G ′ is completely solvable these cohomologies can be computed in terms of the Lie algebra cohomology. Indeed, it follows from [8] (see also [1]) that H * dR (Λ m \G ′ ) ∼ = H * (g ′ ) and H * θ (Λ m \G ′ ) ∼ = H * θ (g ′ ) for any m ∈ N. According to [17], the k th Betti number of g ′ , β k = dim H k (g ′ ), can be computed in terms of the dimension of Z j (g ′ ) = {α ∈ j g * : dα = 0} as follows:…”

Section: Examples Of Solvmanifolds With Lcs Structures Of the Second mentioning

confidence: 99%

“…A locally conformal symplectic structure (LCS for short) on the manifold M is a non degenerate 2-form ω such that there exists an open cover {U i } and smooth functions f i on U i such that ω i = exp(−f i )ω is a symplectic form on U i . This condition is equivalent to requiring that (1) dω = θ ∧ ω for some closed 1-form θ, called the Lee form. The pair (ω, θ) will be called a LCS structure on M. It is well known that if (ω, θ) is a LCS structure on M, then ω is symplectic if and only if θ = 0.…”

Section: Introductionmentioning

confidence: 99%

“…In last years, special attention has been devoted to the study of left invariant LCS structures on Lie groups (see for instance [1,2,3,4]), with very nice results in the case of LCS structures of the first kind. In this work we focus on LCS structures of the second kind on Lie algebras and solvmanifolds.…”

Section: Introductionmentioning

confidence: 99%

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On a certain class of locally conformal symplectic structures of the second kind

Origlia

2020

Differential Geometry and its Applications

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We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a LCS structure and a suitable representation. Moreover, we characterize all LCS Lie algebras obtained with our construction. Finally, we study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting this kind of structure.2010 Mathematics Subject Classification. 53D05, 22E60, 22E25, 53C15, 22E40.There is another way to distinguish LCS structures, to do this, we can deform the de Rham differential d to obtain the adapted differential operatorfor any differential form α ∈ Ω * (M). Since θ is d-closed, this operator satisfies d 2 θ = 0, thus it defines the Morse-Novikov cohomology H * θ (M) of M relative to the closed 1-form θ (see [13,14]). Note that if θ is exact then H * θ (M) ≃ H * dR (M). It is known that if M is a compact oriented n-dimensional manifold, then H 0 θ (M) = H n θ (M) = 0 for any non exact closed 1-form θ (see for instance [6,7]). For any LCS structure (ω, θ) on M, the 2-form ω defines a cohomology class [ω] θ ∈ H 2 θ (M), since d θ ω = dω − θ ∧ ω = 0. The LCS structure (ω, θ) is said to be exact if ω is d θ -exact or [ω] θ = 0, i.e., ω = dη − θ ∧ η for some 1-form η, and it is non-exact if [ω] θ = 0. It was proved in [18] that if the LCS structure (ω, θ) is of the first kind on M then ω is d θ -exact, i.e., [ω] θ = 0. But the converse is not true. Recently in [3] other cohomologies for LCS manifolds were introduced, inspired by the almost Hermitian setting. More precisely, the authors define the LCS-Bott-Chern cohomology and the LCS-Aeppli cohomology on any compact LCS manifold, and compute them for some LCS solvmanifolds in low dimensions.

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Section: Examples Of Solvmanifolds With Lcs Structures Of the Second mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a certain class of locally conformal symplectic structures of the second kind

Origlia

2020

Differential Geometry and its Applications

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“…In [6] we considered the existence of invariant LCK structures on solvmanifolds associated to almost abelian Lie groups. Firstly, we showed that there are plenty of almost abelian Lie algebras which admit LCK structures.…”

Section: 2mentioning

confidence: 99%

“…However, almost abelian solvmanifolds equipped with invariant LCK structures are very scarce, since we have shown in [6] that they only occur in dimension 4.…”

Section: 2mentioning

confidence: 99%

Locally conformally Kähler solvmanifolds: a survey

Andrada

Origlia

2019

Complex Manifolds

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A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.2010 Mathematics Subject Classification. 53B35, 53A30, 22E25.

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Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

Brienza,

Fino

2024

Mathematische Nachrichten

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In this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of ‐invariant nilradical and non‐‐invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.

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Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

Cited by 34 publications

References 36 publications

On a certain class of locally conformal symplectic structures of the second kind

On a certain class of locally conformal symplectic structures of the second kind

Locally conformally Kähler solvmanifolds: a survey

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

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