Callum R. Brodie, A. Constantin, R. Deen, A. Lukas
Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.
{"title":"Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces","authors":"Callum R. Brodie, A. Constantin, R. Deen, A. Lukas","doi":"10.1515/coma-2020-0115","DOIUrl":"https://doi.org/10.1515/coma-2020-0115","url":null,"abstract":"Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"223 - 229"},"PeriodicalIF":0.5,"publicationDate":"2019-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43123197","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 a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit representations of the Lie algebras 𝔰𝔭𝔦𝔶 (8), 𝔰𝔭𝔦𝔶 (7) and 𝔤2, etc.
{"title":"A binary encoding of spinors and applications","authors":"Gerardo Arizmendi, R. Herrera","doi":"10.1515/coma-2020-0100","DOIUrl":"https://doi.org/10.1515/coma-2020-0100","url":null,"abstract":"Abstract We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit representations of the Lie algebras 𝔰𝔭𝔦𝔶 (8), 𝔰𝔭𝔦𝔶 (7) and 𝔤2, etc.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"162 - 193"},"PeriodicalIF":0.5,"publicationDate":"2019-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45775957","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 show that a 2k-current T on a complex manifold is a real holomorphic k-chain if and only if T is locally real rectifiable, d-closed and has ℋ2k-locally finite support. This result is applied to study homology classes represented by algebraic cycles.
{"title":"A characterization of real holomorphic chains and applications in representing homology classes by algebraic cycles","authors":"J. Teh, Chin-Jui Yang","doi":"10.1515/coma-2020-0005","DOIUrl":"https://doi.org/10.1515/coma-2020-0005","url":null,"abstract":"Abstract We show that a 2k-current T on a complex manifold is a real holomorphic k-chain if and only if T is locally real rectifiable, d-closed and has ℋ2k-locally finite support. This result is applied to study homology classes represented by algebraic cycles.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"7 1","pages":"93 - 105"},"PeriodicalIF":0.5,"publicationDate":"2019-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49582500","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}
J. Dorfmeister, Walter Freyn, Shimpei Kobayashi, Erxiao Wang
Abstract The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8]. In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.
{"title":"Survey on real forms of the complex A2(2)-Toda equation and surface theory","authors":"J. Dorfmeister, Walter Freyn, Shimpei Kobayashi, Erxiao Wang","doi":"10.1515/coma-2019-0011","DOIUrl":"https://doi.org/10.1515/coma-2019-0011","url":null,"abstract":"Abstract The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8]. In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"194 - 227"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47632782","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 classical theoremof Kervaire states that products of spheres are parallelizable if and only if at least one of the factors has odd dimension. Two explicit parallelizations on Sm × S2h−1 seem to be quite natural, and have been previously studied by the first named author in [32]. The present paper is devoted to the three choices G = G2, Spin(7), Spin(9) of G-structures on Sm × S2h−1, respectively with m + 2h − 1 = 7, 8, 16 and related with octonionic geometry.
{"title":"Parallelizations on products of spheres and octonionic geometry","authors":"M. Parton, P. Piccinni","doi":"10.1515/coma-2019-0007","DOIUrl":"https://doi.org/10.1515/coma-2019-0007","url":null,"abstract":"Abstract A classical theoremof Kervaire states that products of spheres are parallelizable if and only if at least one of the factors has odd dimension. Two explicit parallelizations on Sm × S2h−1 seem to be quite natural, and have been previously studied by the first named author in [32]. The present paper is devoted to the three choices G = G2, Spin(7), Spin(9) of G-structures on Sm × S2h−1, respectively with m + 2h − 1 = 7, 8, 16 and related with octonionic geometry.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"138 - 149"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48226626","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 For a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space form M is weakly locally pseudo-Hermitian symmetric if and only if (i) dim M = 3, (ii) M is a Sasakian space form, or (iii) M is locally isometric to the unit tangent sphere bundle T1(n+1) of a hyperbolic space n+1 of constant curvature −1.
摘要对于接触流形，研究了Tanaka-Webster连接下具有常全纯截面曲率的强伪凸CR空间形式。我们证明了强伪凸CR空间形式M是弱局部伪埃米对称的当且仅当(i) dim M = 3， (ii) M是Sasakian空间形式，或(iii) M局部等距于恒定曲率- 1的双曲空间n+1的单位切线球束T1(n+1)。
{"title":"Strongly pseudo-convex CR space forms","authors":"Jong Taek Cho","doi":"10.1515/coma-2019-0014","DOIUrl":"https://doi.org/10.1515/coma-2019-0014","url":null,"abstract":"Abstract For a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space form M is weakly locally pseudo-Hermitian symmetric if and only if (i) dim M = 3, (ii) M is a Sasakian space form, or (iii) M is locally isometric to the unit tangent sphere bundle T1(n+1) of a hyperbolic space n+1 of constant curvature −1.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"279 - 293"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44770620","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 The Gauss images of isoparametric hypersufaces of the standard sphere Sn+1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Qn(ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.
{"title":"Lagrangian geometry of the Gauss images of isoparametric hypersurfaces in spheres","authors":"R. Miyaoka, Y. Ohnita","doi":"10.1515/coma-2019-0013","DOIUrl":"https://doi.org/10.1515/coma-2019-0013","url":null,"abstract":"Abstract The Gauss images of isoparametric hypersufaces of the standard sphere Sn+1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Qn(ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"265 - 278"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41377214","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 note we show families of homogeneous Sasakian manifolds G/H which are nonformal. The non-formality condition is expressed in terms of characters of a maximal torus in G.
摘要本文给出了齐次Sasakian流形G/H的非正规族。非形式条件用G中极大环面的性质来表示。
{"title":"On formality of homogeneous Sasakian manifolds","authors":"Irena Morocka-Tralle, A. Tralle","doi":"10.1515/coma-2019-0009","DOIUrl":"https://doi.org/10.1515/coma-2019-0009","url":null,"abstract":"Abstract In this note we show families of homogeneous Sasakian manifolds G/H which are nonformal. The non-formality condition is expressed in terms of characters of a maximal torus in G.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"160 - 169"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42045572","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 survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.
{"title":"A Survey of Riemannian Contact Geometry","authors":"D. Blair","doi":"10.1515/coma-2019-0002","DOIUrl":"https://doi.org/10.1515/coma-2019-0002","url":null,"abstract":"Abstract This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"31 - 64"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47447073","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 discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number (n+1)(n+2)2 ${{(n + 1)(n + 2)} over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.
{"title":"Contact metric manifolds with large automorphism group and (κ, µ)-spaces","authors":"A. Lotta","doi":"10.1515/coma-2019-0015","DOIUrl":"https://doi.org/10.1515/coma-2019-0015","url":null,"abstract":"Abstract We discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number (n+1)(n+2)2 ${{(n + 1)(n + 2)} over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"6 1","pages":"294 - 302"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2019-0015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43510799","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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