几何的对数形式和复杂结构的变形

IF 0.9 1区 数学 Q2 MATHEMATICS Journal of Algebraic Geometry Pub Date : 2017-07-31 DOI:10.1090/JAG/723
Kefeng Liu, S. Rao, Xueyuan Wan
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引用次数: 17

摘要

我们提出了一种求解某∂¯的新方法 \bar \partial -利用谐波积分理论求解Kähler流形上电流的对数微分形式方程。结果可以看作是∂∂¯ \partial \bar \partial 对数形式的引理。作为应用,我们推广了Deligne关于对数形式的闭性的结果,给出了Deligne关于对数Hodge的简并定理在e1e_1能级上对de Rham谱序列的几何证明和更简单的证明,以及紧形Kähler流形上的一个注入定理。此外,对于Kähler流形上的一类复杂结构的对数变形,我们构造了在中心纤维上任意对数(n,q) (n,q) -形式的可拓,从而通过将迭代方法推广到对数形式,推导出对数Calabi-Yau结构的局部稳定性。最后用微分几何方法证明了对数Calabi-Yau对和Calabi-Yau流形上的一对变形的无障碍性。
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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.
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来源期刊
CiteScore
2.70
自引率
5.60%
发文量
23
审稿时长
>12 weeks
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.
期刊最新文献
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