Abstract The purpose in this paper is to construct solvmanifolds without LCK structures such that the complex structure is left-invariant
摘要本文的目的是构造没有LCK结构的溶剂流形，使复结构保持不变
{"title":"Examples of solvmanifolds without LCK structures","authors":"H. Sawai","doi":"10.1515/coma-2018-0005","DOIUrl":"https://doi.org/10.1515/coma-2018-0005","url":null,"abstract":"Abstract The purpose in this paper is to construct solvmanifolds without LCK structures such that the complex structure is left-invariant","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"5 1","pages":"103 - 110"},"PeriodicalIF":0.5,"publicationDate":"2018-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2018-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46608071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study special Lagrangian fibrations of SU(3)-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group G, we decompose such SU(3)-structures into triples of solder 1-forms, connection 1-forms and equivariant 3 × 3 positive-definite symmetric matrix-valued functions on principal G-bundles over 3-manifolds. As applications, we describe regular parts of G2-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T3 and SO(3).
{"title":"G2-metrics arising from non-integrable special Lagrangian fibrations","authors":"Ryohei Chihara","doi":"10.1515/coma-2019-0019","DOIUrl":"https://doi.org/10.1515/coma-2019-0019","url":null,"abstract":"Abstract We study special Lagrangian fibrations of SU(3)-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group G, we decompose such SU(3)-structures into triples of solder 1-forms, connection 1-forms and equivariant 3 × 3 positive-definite symmetric matrix-valued functions on principal G-bundles over 3-manifolds. As applications, we describe regular parts of G2-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T3 and SO(3).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"348 - 365"},"PeriodicalIF":0.5,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44815241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniele Angella, T. Suwa, Nicoletta Tardini, A. Tomassini
Abstract We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.
{"title":"Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms","authors":"Daniele Angella, T. Suwa, Nicoletta Tardini, A. Tomassini","doi":"10.1515/coma-2020-0103","DOIUrl":"https://doi.org/10.1515/coma-2020-0103","url":null,"abstract":"Abstract We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"194 - 214"},"PeriodicalIF":0.5,"publicationDate":"2017-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0103","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44478161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a > [0,∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.
{"title":"A Family of Complex Nilmanifolds with in finitely Many Real Homotopy Types","authors":"A. Latorre, L. Ugarte, R. Villacampa","doi":"10.1515/coma-2018-0004","DOIUrl":"https://doi.org/10.1515/coma-2018-0004","url":null,"abstract":"Abstract We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a > [0,∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"5 1","pages":"102 - 89"},"PeriodicalIF":0.5,"publicationDate":"2017-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2018-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43188185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.
{"title":"A survey on Inverse mean curvature flow in ROSSes","authors":"Giuseppe Pipoli","doi":"10.1515/coma-2017-0016","DOIUrl":"https://doi.org/10.1515/coma-2017-0016","url":null,"abstract":"Abstract In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"245 - 262"},"PeriodicalIF":0.5,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44411308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this short note, we prove that a Calabi extremal Kähler-Ricci soliton on a compact toric Kähler manifold is Einstein. This settles for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.
{"title":"Toric extremal Kähler-Ricci solitons are Kähler-Einstein","authors":"Simone Calamai, David Petrecca","doi":"10.1515/coma-2017-0012","DOIUrl":"https://doi.org/10.1515/coma-2017-0012","url":null,"abstract":"Abstract In this short note, we prove that a Calabi extremal Kähler-Ricci soliton on a compact toric Kähler manifold is Einstein. This settles for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"179 - 182"},"PeriodicalIF":0.5,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49594150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation p: π1 (S) → PSL2ℝ x PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h' on S induced by the action of p on ℍ2 x ℍ2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h’).
抽象给出负曲率的表面光滑spacelike∑反德西特空间维度3,不变的表示p:π- 1 (S)→PSL2ℝx PSL2ℝ面向,S是一个封闭的表面属≥2,规范化建设associates∑微分同胚映射φ∑的美国原来φ∑是该地区的symplectomorphism形式的两个双曲指标h, h p的行动引起的在年代ℍ2 xℍ2。利用与通量同态有关的一个代数构造，给出了φ∑是(S, h)的哈密顿辛同态与(S, h)到(S, h’)的唯一极小拉格朗日微分同态的复合的一个新的证明。
{"title":"The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry","authors":"Andrea Seppi","doi":"10.1515/coma-2017-0013","DOIUrl":"https://doi.org/10.1515/coma-2017-0013","url":null,"abstract":"Abstract Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation p: π1 (S) → PSL2ℝ x PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h' on S induced by the action of p on ℍ2 x ℍ2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h’).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"183 - 199"},"PeriodicalIF":0.5,"publicationDate":"2017-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47444633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.
{"title":"Kähler metrics via Lorentzian Geometry in dimension four","authors":"A. Aazami, G. Maschler","doi":"10.1515/coma-2020-0002","DOIUrl":"https://doi.org/10.1515/coma-2020-0002","url":null,"abstract":"Abstract Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"36 - 61"},"PeriodicalIF":0.5,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44282245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.
{"title":"Kähler-Einstein metrics: Old and New","authors":"Daniele Angella, Cristiano Spotti","doi":"10.1515/coma-2017-0014","DOIUrl":"https://doi.org/10.1515/coma-2017-0014","url":null,"abstract":"Abstract We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course \"Kähler-Einstein metrics\" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"200 - 244"},"PeriodicalIF":0.5,"publicationDate":"2017-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48411379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.
摘要我们研究紧连通黎曼曲面上全纯主丛上的连通，以及更普遍的对数连通。
{"title":"Criterion for connections on principal bundles over a pointed Riemann surface","authors":"I. Biswas","doi":"10.1515/coma-2017-0010","DOIUrl":"https://doi.org/10.1515/coma-2017-0010","url":null,"abstract":"Abstract We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"155 - 171"},"PeriodicalIF":0.5,"publicationDate":"2017-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44685567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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