Non-Kähler solvmanifolds with generalized Kähler structure

Fino, Anna; Томассини, Адриано

doi:10.4310/jsg.2009.v7.n2.a1

Cited by 32 publications

(51 citation statements)

References 26 publications

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“…Together with Theorem , this yields new examples of invariant generalized Kähler structures on Lie groups. They are all however direct generalizations of the examples provided in . Corollary There exists a simply‐connected, almost‐abelian Lie group

(G, J)

with left‐invariant complex structure

J

, on which the pluriclosed flow of left‐invariant metrics has some solutions with finite extinction time and some that exist for all positive times.…”

Section: Introductionmentioning

confidence: 91%

“…Example In [, § 3], the authors study a 6‐dimensional, almost‐abelian solvable Lie algebra

s_{a, b}

which according to Notation may be described as

(R^{6}, μ false(a, v, A false))

, with

v = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}), A = (\begin{matrix} - 0false \frac{a}{2} & 0 & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & - b & 0 & 0 \\ 0 & 0 & 0 & - 0false \frac{a}{2} \end{matrix}), a, b \in double-struckR ∖ {0} .

Notice that we have rearranged the basis for consistency within this article. The complex structure is given by

J e_{1} = e_{6}

J e_{2} = e_{5}

J e_{3} = e_{4}

; thus by Lemma , it defines an integrable complex structure on the corresponding simply‐connected Lie group

S_{a, b}

.…”

Section: Almost‐abelian Solvmanifoldsmentioning

confidence: 99%

“…A Lie group is called almost‐abelian if its Lie algebra has a codimension‐one abelian ideal. In spite of having a simple Lie‐theoretical description, its compact quotients provide a rich family of examples of compact complex manifolds, including for instance hyperelliptic surfaces, Inoue surfaces of type

S^{0}

, primary Kodaira surfaces (these are also quotients of 2‐step nilpotent Lie groups), and Fino and Tomassini's 6‐dimensional solvmanifold admitting a generalized Kähler structure but no Kähler structures , just to name a few.…”

Section: Introductionmentioning

confidence: 99%

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The long‐time behavior of the homogeneous pluriclosed flow

Arroyo

Lafuente

2019

Proc. London Math. Soc.

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We study the asymptotic behavior of the pluriclosed flow in the case of left‐invariant Hermitian structures on Lie groups. We prove that solutions on 2‐step nilpotent Lie groups and on almost‐abelian Lie groups converge, after a suitable normalization, to self‐similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

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(G, J)

with left‐invariant complex structure

J

, on which the pluriclosed flow of left‐invariant metrics has some solutions with finite extinction time and some that exist for all positive times.…”

Section: Introductionmentioning

confidence: 91%

“…Example In [, § 3], the authors study a 6‐dimensional, almost‐abelian solvable Lie algebra

s_{a, b}

which according to Notation may be described as

(R^{6}, μ false(a, v, A false))

, with

v = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}), A = (\begin{matrix} - 0false \frac{a}{2} & 0 & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & - b & 0 & 0 \\ 0 & 0 & 0 & - 0false \frac{a}{2} \end{matrix}), a, b \in double-struckR ∖ {0} .

Notice that we have rearranged the basis for consistency within this article. The complex structure is given by

J e_{1} = e_{6}

J e_{2} = e_{5}

J e_{3} = e_{4}

; thus by Lemma , it defines an integrable complex structure on the corresponding simply‐connected Lie group

S_{a, b}

.…”

Section: Almost‐abelian Solvmanifoldsmentioning

confidence: 99%

S^{0}

Section: Introductionmentioning

confidence: 99%

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The long‐time behavior of the homogeneous pluriclosed flow

Arroyo

Lafuente

2019

Proc. London Math. Soc.

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“…. , a 12 and define an integrable almost complex structure J on n. Let N be the simply connected Lie group with Lie algebra n. Then, for any a 1 , . .…”

Section: -Dimensional Nilmanifoldsmentioning

confidence: 99%

“…Strong KT metrics have been recently studied by many authors and they have also applications in type II string theory and in 2-dimensional supersymmetric σ-models [15,23,39]. Moreover, they have also links with generalized Kähler structures (see, for instance, [2,11,12,15,19,22]). New simply connected strong KT examples have been recently constructed by Swann [40] via the twist construction, by reproducing the 6-dimensional examples found previously in [18].…”

Section: Introductionmentioning

confidence: 99%

On astheno-Kähler metrics

Fino

Томассини

2010

Journal of the London Mathematical Society

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A Hermitian metric on a complex manifold of complex dimension n is called astheno-Kähler if its fundamental 2-form F satisfies the condition ∂∂F n−2 = 0. If n = 3, then the metric is strong KT, that is, F is ∂∂-closed. By using blow-ups and the twist construction, we construct simply connected astheno-Kähler manifolds of complex dimension n > 3. Moreover, we construct a family of astheno-Kähler (non-strong KT) 2-step nilmanifolds of complex dimension 4 and we study deformations of strong KT structures on nilmanifolds of complex dimension 3. Finally, we study the relation between the astheno-Kähler condition and the (locally) conformally balanced condition and we provide examples of locally conformally balanced astheno-Kähler metrics on T 2 -bundles over (non-Kähler) homogeneous complex surfaces.

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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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Non-Kähler solvmanifolds with generalized Kähler structure

Cited by 32 publications

References 26 publications

The long‐time behavior of the homogeneous pluriclosed flow

The long‐time behavior of the homogeneous pluriclosed flow

On astheno-Kähler metrics

Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

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