GEOMETRY OF LOGARITHMIC FORMS AND DEFORMATIONS OF COMPLEX STRUCTURES
Journal article, 2019

We present a new method to solve certain (partial derivative) over bar -equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a partial derivative(partial derivative) over bar -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-level, as well as a certain injectivity theorem on compact Kahler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kahler manifolds, we construct the extension for any logarithmic (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.

Author

Kefeng Liu

Chongqing Technology and Business University

University of California at Los Angeles

Sheng Rao

Wuhan University

Xueyuan Wan

Chalmers, Mathematical Sciences, Algebra and geometry

University of Gothenburg

Journal of Algebraic Geometry

1056-3911 (ISSN) 1534-7486 (eISSN)

Vol. 28 4 773-815

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1090/jag/723

More information

Latest update

1/30/2020