Balanced Hermitian structures on almost abelian Lie algebras

Fino, Anna; Paradiso, Fabio

doi:10.48550/arxiv.2011.09992

Cited by 6 publications

(21 citation statements)

References 41 publications

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“…We stress that, unimodular solvable Lie groups of dimension 6 admitting nowhere vanishing holomorphic (3, 0)-forms have been classified in [19]. Furthermore, the results of Proposition 2.1 and Proposition 2.2 have also been obtained by Fino and Paradiso in [20], using different techniques.…”

Section: Almost-abelian Lie Groupsmentioning

confidence: 86%

“…In the latter case, it was proved that the Anomaly flow preserves the balanced and the locally conformally Kähler conditions for any couple of Gauduchon connections ∇ τ and ∇, and explicit solutions to the flow were found, both with 'flat' and 'non-flat' bundle. Finally, some results on the Anomaly flow over almost-abelian Lie groups were recently obtained by Fino and Paradiso in [20].…”

Section: Introductionmentioning

confidence: 94%

“…Let µ 0 = µ 0 (0, 0, A 0 ) be an almost-abelian balanced Lie bracket on (g, J). Then, the bracket flow (20) starting at µ 0 reduces to…”

mentioning

confidence: 99%

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The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Pujia

2021

Preprint

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We study the Hull-Strominger system and the Anomaly flow on a special class of 2-step solvmanifolds, namely the class of almost-abelian Lie groups. In this setting, we characterize the existence of invariant solutions to the Hull-Strominger system with respect to the family of Gauduchon connections in the anomaly cancellation equation. Then, motivated by the results on the Anomaly flow, we investigate the flow of invariant metrics in our setting, proving that it always reduces to a flow of a special form. Finally, under an extra assumption on the initial metrics, we show that the flow is immortal and, when the slope parameter is zero, it always converges to a Kähler metric

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Section: Almost-abelian Lie Groupsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 94%

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The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Pujia

2021

Preprint

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“…We recall the following Lemma 6.1. ( [18]) The Lee form of a Hermitian Lie algebra (g, J, g) satisfies…”

Section: Balanced Hermitian Structuresmentioning

confidence: 99%

Hermitian structures on a class of almost nilpotent solvmanifolds

Fino¹,

Paradiso²

2021

Preprint

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In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients Γ\G, where G is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and Γ is a lattice of G. We first obtain a characterization of Hermitian almost nilpotent Lie algebras g whose nilradical n has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds.Finally, we prove some non-existence results for generalized Kähler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized Kähler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-Kähler) manifolds admitting non-split generalized Kähler structures.

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“…Kähler, SKT, balanced and LCK almost abelian Lie algebras were studied in terms of the data (a, v, A) in [23,7,14,15,2]. Six-dimensional almost abelian Lie algebras admitting SKT structures were classified in [14], and in [19] the result was extended to a wider class of two-step solvable Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

Locally conformally balanced metrics on almost abelian Lie algebras

Paradiso¹

2021

Preprint

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We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

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Balanced Hermitian structures on almost abelian Lie algebras

Cited by 6 publications

References 41 publications

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds

Hermitian structures on a class of almost nilpotent solvmanifolds

Locally conformally balanced metrics on almost abelian Lie algebras

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