{"title":"几何的对数形式和复杂结构的变形","authors":"Kefeng Liu, S. Rao, Xueyuan Wan","doi":"10.1090/JAG/723","DOIUrl":null,"url":null,"abstract":"We present a new method to solve certain \n\n \n \n \n ∂\n ¯\n \n \n \\bar \\partial\n \n\n-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a \n\n \n \n ∂\n \n \n ∂\n ¯\n \n \n \n \\partial \\bar \\partial\n \n\n-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at \n\n \n \n E\n 1\n \n E_1\n \n\n-level, as well as a certain injectivity theorem on compact Kähler manifolds.\n\nFurthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic \n\n \n \n (\n n\n ,\n q\n )\n \n (n,q)\n \n\n-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2017-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/723","citationCount":"17","resultStr":"{\"title\":\"Geometry of logarithmic forms and deformations of complex structures\",\"authors\":\"Kefeng Liu, S. Rao, Xueyuan Wan\",\"doi\":\"10.1090/JAG/723\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We present a new method to solve certain \\n\\n \\n \\n \\n ∂\\n ¯\\n \\n \\n \\\\bar \\\\partial\\n \\n\\n-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a \\n\\n \\n \\n ∂\\n \\n \\n ∂\\n ¯\\n \\n \\n \\n \\\\partial \\\\bar \\\\partial\\n \\n\\n-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at \\n\\n \\n \\n E\\n 1\\n \\n E_1\\n \\n\\n-level, as well as a certain injectivity theorem on compact Kähler manifolds.\\n\\nFurthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic \\n\\n \\n \\n (\\n n\\n ,\\n q\\n )\\n \\n (n,q)\\n \\n\\n-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2017-07-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1090/JAG/723\",\"citationCount\":\"17\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/JAG/723\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/JAG/723","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Geometry of logarithmic forms and deformations of complex structures
We present a new method to solve certain
∂
¯
\bar \partial
-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a
∂
∂
¯
\partial \bar \partial
-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at
E
1
E_1
-level, as well as a certain injectivity theorem on compact Kähler manifolds.
Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic
(
n
,
q
)
(n,q)
-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.
期刊介绍:
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.