Abstract We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.
{"title":"Deformation classes in generalized Kähler geometry","authors":"Matthew Gibson, J. Streets","doi":"10.1515/coma-2020-0101","DOIUrl":"https://doi.org/10.1515/coma-2020-0101","url":null,"abstract":"Abstract We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"241 - 256"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41434249","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 a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length. Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.
{"title":"Pseudo-holomorphic curves: A very quick overview","authors":"Gonçalo Oliveira","doi":"10.1515/coma-2020-0105","DOIUrl":"https://doi.org/10.1515/coma-2020-0105","url":null,"abstract":"Abstract This is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length. Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"215 - 229"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0105","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46186694","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 Serre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies dimE1p,q=dimE1n−p,n−q dim E_1^{p,q} = dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” dimEkp,q=dimEkn−p,n−q dim E_k^{p,q} = dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].
{"title":"Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence","authors":"A. Milivojević","doi":"10.1515/coma-2020-0008","DOIUrl":"https://doi.org/10.1515/coma-2020-0008","url":null,"abstract":"Abstract Serre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies dimE1p,q=dimE1n−p,n−q dim E_1^{p,q} = dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” dimEkp,q=dimEkn−p,n−q dim E_k^{p,q} = dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"141 - 144"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47906922","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 Let M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let ω̄ : 𝒜̂ → M be the relative Albanese over M. We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over M.
{"title":"Complex Lagrangians in a hyperKähler manifold and the relative Albanese","authors":"I. Biswas, T. G'omez, André G. Oliveira","doi":"10.1515/coma-2020-0106","DOIUrl":"https://doi.org/10.1515/coma-2020-0106","url":null,"abstract":"Abstract Let M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let ω̄ : 𝒜̂ → M be the relative Albanese over M. We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over M.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"230 - 240"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0106","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42449926","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 introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B i¯j¯λBλji≥0 B_{bar ibar j}^lambda B_{lambda j}^i ge 0 , then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.
摘要引入了近似厄米流形上的Kähler-like、G-Kähler-like度量。我们证明了一个紧致的Kähler-like和G-Kähler-like几乎厄米流形具有一个几乎平衡度量是Kähler。我们还证明了如果一个Kähler-like和G-Kähler-like几乎厄米流形满足B i¯j¯λBλji≥0 B＿ {bar i bar j}^lambda B＿ {lambda j}^ige 0，则度量是几乎平衡的，并且几乎复杂的结构是可积的，这意味着度量是平衡的。我们研究了在某些假设下的G-Kähler-like概厄米流形。
{"title":"On Kähler-like and G-Kähler-like almost Hermitian manifolds","authors":"Masaya Kawamura","doi":"10.1515/coma-2020-0009","DOIUrl":"https://doi.org/10.1515/coma-2020-0009","url":null,"abstract":"Abstract We introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B i¯j¯λBλji≥0 B_{bar ibar j}^lambda B_{lambda j}^i ge 0 , then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"145 - 161"},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47193658","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 paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.
{"title":"On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree","authors":"John A. Arredondo, Camilo Ramírez Maluendas","doi":"10.1515/coma-2020-0004","DOIUrl":"https://doi.org/10.1515/coma-2020-0004","url":null,"abstract":"Abstract In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"73 - 92"},"PeriodicalIF":0.5,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45388734","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 paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C0−closed in the group Diff∞ (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C0−limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.
{"title":"On Cosymplectic Dynamics I","authors":"S. Tchuiaga, F. Houenou, P. Bikorimana","doi":"10.1515/coma-2021-0132","DOIUrl":"https://doi.org/10.1515/coma-2021-0132","url":null,"abstract":"Abstract This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C0−closed in the group Diff∞ (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C0−limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogue of a result that was proved by Hofer-Zehnder follows.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"114 - 137"},"PeriodicalIF":0.5,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47386855","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 prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.
{"title":"Kobayashi—Hitchin correspondence for twisted vector bundles","authors":"A. Perego","doi":"10.1515/coma-2020-0107","DOIUrl":"https://doi.org/10.1515/coma-2020-0107","url":null,"abstract":"Abstract We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"1 - 95"},"PeriodicalIF":0.5,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0107","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49262617","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 consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.
{"title":"Differential operators on almost-Hermitian manifolds and harmonic forms","authors":"Nicoletta Tardini, A. Tomassini","doi":"10.1515/coma-2020-0006","DOIUrl":"https://doi.org/10.1515/coma-2020-0006","url":null,"abstract":"Abstract We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"106 - 128"},"PeriodicalIF":0.5,"publicationDate":"2019-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588231","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 paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality χ(Tf∑)+χ(Nf∑)≤±c1(f*T(1,0)M). chi left( {{T_f}sum } right) + chi left( {{N_f}sum } right) le pm {c_1}left( {f*{T^{left( {1,0} right)}}M} right). The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.
{"title":"The Adjunction Inequality for Weyl-Harmonic Maps","authors":"Robert Ream","doi":"10.1515/coma-2020-0007","DOIUrl":"https://doi.org/10.1515/coma-2020-0007","url":null,"abstract":"Abstract In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality χ(Tf∑)+χ(Nf∑)≤±c1(f*T(1,0)M). chi left( {{T_f}sum } right) + chi left( {{N_f}sum } right) le pm {c_1}left( {f*{T^{left( {1,0} right)}}M} right). The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"129 - 140"},"PeriodicalIF":0.5,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48553432","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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