Compact complex surfaces with geometric structures related to split quaternions

Davidov, Johann; Grantcharov, Gueo; Мушкаров, Олег; Yotov, Mirroslav

doi:10.1016/j.nuclphysb.2012.07.024

Cited by 10 publications

(9 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely, such a structure determines a conformal class of compatible metrics up to a double cover of M [6]. Examples of almost para-hypercomplex structures that do not admit compatible metrics are given in [6,7].…”

Section: Para-hyperhermitian Structuresmentioning

confidence: 99%

“…These structures appear in [18] as models for superstring theory with N=2 supersymmetry and [11] in relation to deformation spaces of harmonic maps from Riemann surfaces into Lie groups. Note also that such structures have been used recently by B. Klingler [15] in his proof of the Chern conjecture for affine manifolds.In our previous papers [6,7] we initiated the study of compact para-hyperhermitian surfaces, which are the neutral analog of the hyperhermitian 4-manifolds. These surfaces are anti-self-dual as in the positive definite case, but in contrast to the well-known classification of compact hyperhermitian surfaces [4], there are many more compact examples of para-hyperhermitian surfaces.…”

mentioning

confidence: 99%

“…In our previous papers [6,7] we initiated the study of compact para-hyperhermitian surfaces, which are the neutral analog of the hyperhermitian 4-manifolds. These surfaces are anti-self-dual as in the positive definite case, but in contrast to the well-known classification of compact hyperhermitian surfaces [4], there are many more compact examples of para-hyperhermitian surfaces.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Complex surfaces and null conformal Killing vector fields

Davidov¹,

Grantcharov²,

Мушкаров³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of signature (2, 2). We establish first the topological types of pseudo-Hermitian surfaces admitting a nowhere vanishing null vector field. Then we show that a pair of orthogonal, pointwise linearly independent, null, conformal Killing vector fields defines a para-hyperhermitian structure and use this fact for a classification of the smooth compact four-manifolds admitting such a pair of vector fields. We also provide examples of neutral metrics with two orthogonal, pointwise linearly independent, null Killing vector fields on most of these manifolds.There has also been some interest in compact 4-manifolds admitting neutral metrics and compatible complex or para-hypercomplex structures since topological information like the Kodaira classification of compact complex surfaces allows one to study global properties. Important results in this direction are the classifications of compact pseudo-Kähler Einstein and para-hyperkähler surfaces obtained by Petean [21] and Kamada [14], respectively. These structures appear in [18] as models for superstring theory with N=2 supersymmetry and [11] in relation to deformation spaces of harmonic maps from Riemann surfaces into Lie groups. Note also that such structures have been used recently by B. Klingler [15] in his proof of the Chern conjecture for affine manifolds.In our previous papers [6,7] we initiated the study of compact para-hyperhermitian surfaces, which are the neutral analog of the hyperhermitian 4-manifolds. These surfaces are anti-self-dual as in the positive definite case, but in contrast to the well-known classification of compact hyperhermitian surfaces [4], there are many more compact examples of para-hyperhermitian surfaces. In our study, we noticed also that any two pointwise independent, null, orthogonal and parallel vector fields on a 4-manifold with a neutral metric define a para-hyperkäher structure and conversely, any compact para-hyperkäher surface admits two vector fields with the above properties [14]. In this note, we weaken these conditions and study the neutral Hermitian surfaces admitting non-vanishing null conformal Killing vector fields. We observe first that the existence of a non-vanishing null vector field on a compact neutral Hermitian surface leads to a topological restriction which implies a rough classification of these surfaces. Next, we show that the existence of two point-wise independent, null and orthogonal conformal Killing vector fields leads to the existence of a para-hyperhermitian structure. These surfaces were studied in [6, 7] and here we consider the problem for existence of two Killing vector fields having the above properties with respect to the corresponding metrics. Now we describe shortly the content of the paper. In Section 2, some known facts about almost para-hyperhermitian structures are collected and in Lemma 2 they are c...

show abstract

Section: Para-hyperhermitian Structuresmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Complex surfaces and null conformal Killing vector fields

Davidov¹,

Grantcharov²,

Мушкаров³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The para-hypercomplex structure (I, J, K) we defined on R 4 nonetheless descends to X. Let For other examples of para-hyperHermitian structures that are not para-hyperKähler, we refer the reader to [47,48] and the references therein.…”

Section: Para-hyperhermitian and Born Structures 59mentioning

confidence: 99%

Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry

Hu,

Moraru,

Svoboda

2019

Preprint

View full text Add to dashboard Cite

In this paper, we study the geometries given by commuting pairs of generalized endomorphisms A ∈ End(T ⊕ T * ) with the property that their product defines a generalized metric. There are four types of such commuting pairs: generalized Kähler (GK), generalized para-Kähler (GpK), generalized chiral and generalized anti-Kähler geometries. We show that GpK geometry is equivalent to a pair of para-Hermitian structures and we derive the integrability conditions in terms of these. From the physics point of view, this is the geometry of 2D (2, 2) twisted supersymmetric sigma models. The generalized chiral structures are equivalent to a pair of tangent bundle product structures that also appear in physics applications of 2D sigma models. We show that the case when the two product structures anti-commute corresponds to Born geometry. Lastly, the generalized anti-Kähler structures are equivalent to a pair of anti-Hermitian structures (sometimes called Hermitian with Norden metric). The generalized chiral and anti-Kähler geometries do not have isotropic eigenbundles and therefore do not admit the usual description of integrability in terms of the Dorfman bracket. We therefore use an alternative definition of integrability in terms of the generalized Bismut connection of the corresponding metric, which for GK and GpK commuting pairs recovers the usual integrability conditions and can also be used to define the integrability of generalized chiral and anti-Kähler structures. In addition, it allows for a weakening of the integrability condition, which has various applications in physics.

show abstract

“…Complex product structures on Lie algebras were introduced by Andrada and Salamon in [3]. Lie algebras carrying a complex product structure are closely related to many important fields in mathematics and mathematical physics, such as Rota-Baxter operators on pre-Lie algebras [11], geometric structures on compact complex surfaces that are related to the split quaternions [7], paraquaternionic Kähler structures [5] and nilpotent Lie algebras [2]. Recently, complex product structures have been extensively investigated in [4,6,19].…”

Section: Introductionmentioning

confidence: 99%

Complex Product Structures on Hom-Lie Algebras

Nourmohammadifar¹,

Peyghan²

2018

Glasgow Math. J.

View full text Add to dashboard Cite

In this paper, we introduce the notion of complex product structures on hom-Lie algebras and show that a hom-Lie algebra carrying a complex product structure is a double hom-Lie algebra and it is also endowed with a hom-left symmetric product. Moreover, we show that a complex product structure on a hom-Lie algebra determines uniquely a left symmetric product such that the complex and the product structures are invariant with respect to it. Finally, we introduce the notion of hyper-para-Kähler hom-Lie algebras and we present an example of hyper-para-Kähler hom-Lie algebras.

show abstract

Compact complex surfaces with geometric structures related to split quaternions

Abstract: openAccessArticle: FalsePage Range: 330-330doi: 10.1016/j.nuclphysb.2012.07.024Harvest Date: 2016-01-12 15:12:15issueName:cover date: 2012-12-11pubType

Cited by 10 publications

References 29 publications

Complex surfaces and null conformal Killing vector fields

Complex surfaces and null conformal Killing vector fields

Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry

Complex Product Structures on Hom-Lie Algebras

Contact Info

Product

Resources

About