Complex Connections with Trivial Holonomy

Andrada, Adrián; Barberis, M. L.; Dotti, Isabel G.

doi:10.1007/978-1-4614-7193-6_2

Cited by 5 publications

(9 citation statements)

References 28 publications

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“…This connection, which was introduced by Lichnerowicz in [20] and appears in the set of canonical Hermitian connections of Gauduchon [17], has torsion tensor of type (1, 1) with respect to the complex structure. In [4], the notion of abelian complex structure on Lie groups was extended to parallelizable manifolds. This generalization amounts to the existence of a complex connection on the complex manifold with trivial holonomy and torsion of type (1, 1) with respect to the complex structure.…”

Section: Introductionmentioning

confidence: 99%

Abelian Hermitian geometry

Andrada

Barberis

Dotti

2012

Differential Geometry and its Applications

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We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.2010 Mathematics Subject Classification. 53C15, 53B35, 53C30.

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Section: Introductionmentioning

confidence: 99%

Abelian Hermitian geometry

Andrada

Barberis

Dotti

2012

Differential Geometry and its Applications

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“…and T ij k := g a k T a ij . HCF is then defined as (1) ∂ t g t = −K(g t ) , g |t=0 = g 0 , where g 0 is a fixed initial Hermitian metric on M and K(g) := S(g) − Q(g) .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1. For a complex unimodular Lie group G, the maximal solution g t to the HCF flow (1) starting at a left-invariant Hermitian metric satisfies d dt g t = −Ric 1,1 (g t ), where Ric(g t ) denotes the Levi-Civita Ricci tensor. The family of left-invariant Hermitian metrics g t is defined for all t ∈ (−ǫ, ∞) for some ǫ > 0, and (1 + t) −1 g t converges as t → ∞ to a non-flat left-invariant HCF soliton ( Ḡ, ḡ), in the Cheeger-Gromov topology.…”

Section: Introductionmentioning

confidence: 99%

“…Let us also mention that other flows in the HCF family have been studied quite extensively in the literature. By HCF family we mean the following: since short-time existence of HCF (1) in the compact case follows from the ellipticity of the operator g → −S(g) acting on the space of Hermitian metrics compatible with a fixed complex structure, it is clear that that by changing the lower order torsion term Q(g) in (1) one also gets a well-defined parabolic evolution equation. In this direction, in [45] Streets and Tian introduced the pluriclosed flow (shortly PCF), a modification of the HCF which preserves the pluriclosed condition ∂ ∂ω = 0, defined by…”

Section: Introductionmentioning

confidence: 99%

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Hermitian curvature flow on unimodular Lie groups and static invariant metrics

Lafuente

Pujia

Vezzoni

2020

Trans. Amer. Math. Soc.

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We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricciflow type equation ∂tgt = −Ric 1,1 (gt). The solution gt always exist for all positive times, and (1 + t) −1 gt converges as t → ∞ in Cheeger-Gromov sense to a non-flat left-invariant soliton (Ḡ,ḡ). Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in [16] for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures.

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“…We denote this connection as ∇ c . This connection can be defined using the following formula [ABD13]:…”

Section: Introductionmentioning

confidence: 99%

On the Hermitian Geometry of $k$-Gauduchon Orthogonal Complex Structures

Khan¹

2018

Preprint

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The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon [Sal95]. His work describes in great detail the role that curvature plays in this question. We instead focus on torsion, which lends itself to somewhat different analysis of the problem. In terms of novel results, we show that for a certain range of k, the spaces of k-Gauduchon orthogonal complex structures are pre-compact within the moduli space of complex structures and give a preliminary picture of how certain special complex structures fit into the space of orthogonal complex structures.

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Complex Connections with Trivial Holonomy

Cited by 5 publications

References 28 publications

Abelian Hermitian geometry

Abelian Hermitian geometry

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

On the Hermitian Geometry of $k$-Gauduchon Orthogonal Complex Structures

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