Abstract An R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.
{"title":"Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces","authors":"Y. Ohnita","doi":"10.1515/coma-2019-0016","DOIUrl":"https://doi.org/10.1515/coma-2019-0016","url":null,"abstract":"Abstract An R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"303 - 319"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48128817","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 some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.
{"title":"On curvature tensors of Norden and metallic pseudo-Riemannian manifolds","authors":"A. Blaga, Antonella Nannicini","doi":"10.1515/coma-2019-0008","DOIUrl":"https://doi.org/10.1515/coma-2019-0008","url":null,"abstract":"Abstract We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"150 - 159"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46850043","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 describe the natural geometry of Hilbert schemes of curves in ℙ3and, in some cases, in ℙn, n ≥ 4.
摘要我们在中描述了曲线的Hilbert格式的自然几何ℙ3在某些情况下ℙn、 n≥4。
{"title":"Differential geometry of Hilbert schemes of curves in a projective space","authors":"R. Bielawski, Carolin Peternell","doi":"10.1515/coma-2019-0018","DOIUrl":"https://doi.org/10.1515/coma-2019-0018","url":null,"abstract":"Abstract We describe the natural geometry of Hilbert schemes of curves in ℙ3and, in some cases, in ℙn, n ≥ 4.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"335 - 347"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47549330","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 define a Kähler-like almost Hermitian metric. We will prove that on a compact Kähler-like almost Hermitian manifold (M2n, J, g), if it admits a positive ∂ ̄∂-closed (n − 2, n − 2)-form, then g is a quasi-Kähler metric.
{"title":"On the Kähler-likeness on almost Hermitian manifolds","authors":"Masaya Kawamura","doi":"10.1515/coma-2019-0020","DOIUrl":"https://doi.org/10.1515/coma-2019-0020","url":null,"abstract":"Abstract We define a Kähler-like almost Hermitian metric. We will prove that on a compact Kähler-like almost Hermitian manifold (M2n, J, g), if it admits a positive ∂ ̄∂-closed (n − 2, n − 2)-form, then g is a quasi-Kähler metric.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"366 - 376"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49512297","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}
Beniamino Cappelletti-Montano, Antonio de Nicola, G. Dileo, I. Yudin
Abstract We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.
{"title":"Nearly Sasakian manifolds revisited","authors":"Beniamino Cappelletti-Montano, Antonio de Nicola, G. Dileo, I. Yudin","doi":"10.1515/coma-2019-0017","DOIUrl":"https://doi.org/10.1515/coma-2019-0017","url":null,"abstract":"Abstract We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"320 - 334"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45061050","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 These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby–Wang bundles, that might be useful for other applications in contact topology.
摘要这些是基于2018年6月在撒丁岛卡利亚里举行的RIEMain in Contact会议上的一个迷你课程的笔记。主要主题是Reeb动力学和拓扑之间的联系。讨论的主题包括Reeb流的陷阱、Hamiltonian流的塞子、Weinstein猜想、具有有限个周期轨道的Reeb流以及Reeb流截面的全局表面。重点是构造方法，例如Boothby-Wang束中的接触切割和提升群动作，这可能对接触拓扑中的其他应用有用。
{"title":"Controlled Reeb dynamics — Three lectures not in Cala Gonone","authors":"H. Geiges","doi":"10.1515/coma-2019-0006","DOIUrl":"https://doi.org/10.1515/coma-2019-0006","url":null,"abstract":"Abstract These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby–Wang bundles, that might be useful for other applications in contact topology.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"118 - 137"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44183672","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 This is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.
{"title":"Ricci-flat and Einstein pseudoriemannian nilmanifolds","authors":"D. Conti, F. Rossi","doi":"10.1515/coma-2019-0010","DOIUrl":"https://doi.org/10.1515/coma-2019-0010","url":null,"abstract":"Abstract This is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"170 - 193"},"PeriodicalIF":0.5,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46692702","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 the behavior of the degeneration at the second step of the Frölicher spectral sequence of a 𝒞∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result à la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.
{"title":"On the degeneration of the Frölicher spectral sequence and small deformations","authors":"Michele Maschio","doi":"10.1515/coma-2020-0003","DOIUrl":"https://doi.org/10.1515/coma-2020-0003","url":null,"abstract":"Abstract We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a 𝒞∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result à la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"62 - 72"},"PeriodicalIF":0.5,"publicationDate":"2018-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47681660","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 A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.
{"title":"Locally conformally Kähler solvmanifolds: a survey","authors":"A. Andrada, M. Origlia","doi":"10.1515/coma-2019-0003","DOIUrl":"https://doi.org/10.1515/coma-2019-0003","url":null,"abstract":"Abstract A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"65 - 87"},"PeriodicalIF":0.5,"publicationDate":"2018-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48881027","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 This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.
摘要本文基于在意大利卡利亚里举行的RIEMain in Contact会议上为纪念现代黎曼接触几何创始人之一David Blair 78岁生日所做的一次演讲。本文是对一种特殊类型的黎曼接触结构Sasakian几何的综述。这项调查的最终目标是了解Sasaki结构类的模量，以及极值和常标量曲率Sasaki度量的模量，特别是Sasaki-Enstein度量的模量。
{"title":"Contact Structures of Sasaki Type and Their Associated Moduli","authors":"C. Boyer","doi":"10.1515/coma-2019-0001","DOIUrl":"https://doi.org/10.1515/coma-2019-0001","url":null,"abstract":"Abstract This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"1 - 30"},"PeriodicalIF":0.5,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45179629","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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