Abstract Let (M, D) be a compact contact manifold with dimRM = 2n ≥ 5. This means that: M is a C∞ differential manifold with dimRM = 2n ≥ 5. And D is a subbundle of the tangent bundle TM which satisfying; there is a real one form θ such that D = {X : X ∈ TM, θ(X) = 0}, and θ ^ Λn−1(d ) ≠ 0 at every point of p of M. Especially, we assume that our D admits almost CR structure,(M, S). In this paper, inspired by the work of Matsumoto([M]), we study the difference of partially integrable almost CR structures from actual CR structures. And we discuss partially integrable almost CR structures from the point of view of the deformation theory of CR structures ([A1],[AGL]).
{"title":"Partially integrable almost CR structures","authors":"T. Akahori","doi":"10.1515/coma-2020-0124","DOIUrl":"https://doi.org/10.1515/coma-2020-0124","url":null,"abstract":"Abstract Let (M, D) be a compact contact manifold with dimRM = 2n ≥ 5. This means that: M is a C∞ differential manifold with dimRM = 2n ≥ 5. And D is a subbundle of the tangent bundle TM which satisfying; there is a real one form θ such that D = {X : X ∈ TM, θ(X) = 0}, and θ ^ Λn−1(d ) ≠ 0 at every point of p of M. Especially, we assume that our D admits almost CR structure,(M, S). In this paper, inspired by the work of Matsumoto([M]), we study the difference of partially integrable almost CR structures from actual CR structures. And we discuss partially integrable almost CR structures from the point of view of the deformation theory of CR structures ([A1],[AGL]).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"403 - 414"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43496169","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 an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.
{"title":"A polar dual to the momentum of toric Fano manifolds","authors":"Yuji Sano","doi":"10.1515/coma-2020-0116","DOIUrl":"https://doi.org/10.1515/coma-2020-0116","url":null,"abstract":"Abstract We introduce an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"230 - 246"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0116","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44121386","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 fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.
{"title":"Real rectifiable currents, holomorphic chains and algebraic cycles","authors":"J. Teh, Chin-Jui Yang","doi":"10.1515/coma-2020-0119","DOIUrl":"https://doi.org/10.1515/coma-2020-0119","url":null,"abstract":"Abstract We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"274 - 285"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47259100","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 give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L∞-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.
{"title":"Algebroids, AKSZ Constructions and Doubled Geometry","authors":"V. Marotta, R. Szabo","doi":"10.1515/coma-2020-0125","DOIUrl":"https://doi.org/10.1515/coma-2020-0125","url":null,"abstract":"Abstract We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L∞-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"15 1","pages":"354 - 402"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41273900","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 give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.
摘要本文简要地讨论了严格nef束与充足向量束的异同，重点讨论了严格nef束的射影流形的几何问题。
{"title":"Strictly nef vector bundles and characterizations of ℙn","authors":"Jie Liu, Wenhao Ou, Xiaokui Yang","doi":"10.1515/coma-2020-0109","DOIUrl":"https://doi.org/10.1515/coma-2020-0109","url":null,"abstract":"Abstract In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"148 - 159"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45568258","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 give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.
{"title":"Almost complex manifolds with small Nijenhuis tensor","authors":"L. Fernández, Tobias Shin, Scott O. Wilson","doi":"10.1515/coma-2020-0122","DOIUrl":"https://doi.org/10.1515/coma-2020-0122","url":null,"abstract":"Abstract We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"329 - 335"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49396128","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 𝒯 s,p,n be the canonical blow-up of the Grassmann manifold G(p, n) constructed by blowing up the Plücker coordinate subspaces associated with the parameter s. We prove that the higher cohomology groups of the tangent bundle of 𝒯 s,p,n vanish. As an application, 𝒯s,p,n is locally rigid in the sense of Kodaira-Spencer.
{"title":"A vanishing theorem for the canonical blow-ups of Grassmann manifolds","authors":"Hanlong Fang, Song-Chun Zhu","doi":"10.1515/coma-2020-0126","DOIUrl":"https://doi.org/10.1515/coma-2020-0126","url":null,"abstract":"Abstract Let 𝒯 s,p,n be the canonical blow-up of the Grassmann manifold G(p, n) constructed by blowing up the Plücker coordinate subspaces associated with the parameter s. We prove that the higher cohomology groups of the tangent bundle of 𝒯 s,p,n vanish. As an application, 𝒯s,p,n is locally rigid in the sense of Kodaira-Spencer.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"415 - 439"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47006506","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 Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.
{"title":"Generalized almost even-Clifford manifolds and their twistor spaces","authors":"Luis Fernando Hernández-Moguel, R. Herrera","doi":"10.1515/coma-2020-0108","DOIUrl":"https://doi.org/10.1515/coma-2020-0108","url":null,"abstract":"Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"96 - 124"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0108","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46917540","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 Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ωt }t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {Mt}t.
{"title":"Deformations of Strong Kähler with torsion metrics","authors":"Riccardo Piovani, Tommaso Sferruzza","doi":"10.1515/coma-2020-0120","DOIUrl":"https://doi.org/10.1515/coma-2020-0120","url":null,"abstract":"Abstract Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ωt }t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {Mt}t.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"286 - 301"},"PeriodicalIF":0.5,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42273335","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 S be a compact complex surface in class VII0+ containing a cycle of rational curves C = ∑Dj. Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C′ then C′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj. In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.
{"title":"Non Kählerian surfaces with a cycle of rational curves","authors":"G. Dloussky","doi":"10.1515/coma-2020-0114","DOIUrl":"https://doi.org/10.1515/coma-2020-0114","url":null,"abstract":"Abstract Let S be a compact complex surface in class VII0+ containing a cycle of rational curves C = ∑Dj. Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C′ then C′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj. In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"208 - 222"},"PeriodicalIF":0.5,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48664690","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：