Generalized Kähler almost abelian Lie groups

Fino, Anna; Paradiso, Fabio

doi:10.1007/s10231-020-01059-1

Cited by 26 publications

(31 citation statements)

References 30 publications

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“…associated with ω(t). The flow (7.1) differs from the one introduced in [7] by a factor in the second summand in the right-hand side, as suggested by the authors (see [17]). We shall refer to this flow as balanced flow and we shall denote the right-hand side of (7.2) by q(ω(t)) for simplicity.…”

Section: Balanced Flowmentioning

confidence: 71%

“…In particular, as already described in [17], the bracket flow equation corresponding to the balanced flow is given by (7.4)…”

Section: Bcmentioning

confidence: 99%

“…On this subject, we recall that, by [17], all generalized Kähler structures on 6-dimensional almost abelian Lie algebras are split. In dimension higher than six, this is no longer true, as the next theorem shows.…”

Section: Case (2)mentioning

confidence: 99%

“…SKT structures on special classes of solvable Lie groups were instead studied in [16,5,17,24], with [21,17] focusing on generalized Kähler structures on almost abelian Lie algebras, namely solvable Lie algebras admitting a codimension one abelian ideal. Up to now, in literature, these are the only examples of non-Kähler solvable Lie algebras admitting generalized Kähler structures, with the first example having been found in [21] and a full classification in real dimension six obtained in [17]. Moreover, all the known solvable examples are split.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Balanced Flowmentioning

confidence: 71%

“…In particular, as already described in [17], the bracket flow equation corresponding to the balanced flow is given by (7.4)…”

Section: Bcmentioning

confidence: 99%

Section: Case (2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hermitian structures on a class of almost nilpotent solvmanifolds

Fino¹,

Paradiso²

2021

Preprint

Self Cite

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“…Another possible generalization is given by strong Kähler with torsion (SKT, also known as pluriclosed ) metrics, satisfying ∂∂ω = 0 (see for instance [25,22]). SKT metrics have natural applications in type II string theory and 2-dimensional supersymmetric σ-models [31,51] and are closely related to generalized Kähler structures [33,34,1,24,21]. The balanced and SKT conditions are transversal to each other, in the sense that a Hermitian metric which is balanced and SKT is necessarily Kähler (see [2]).…”

Section: Introductionmentioning

confidence: 99%

Balanced Hermitian structures on almost abelian Lie algebras

Fino¹,

Paradiso²

2020

Preprint

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We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify 6-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [26] that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with leftinvariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [6] and of the anomaly flow [47] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

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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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Generalized Kähler almost abelian Lie groups

Cited by 26 publications

References 30 publications

Hermitian structures on a class of almost nilpotent solvmanifolds

Hermitian structures on a class of almost nilpotent solvmanifolds

Balanced Hermitian structures on almost abelian Lie algebras

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

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