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Journal article
Authors K. F. Liu
S. Rao
Xueyuan Wan
Published in Journal of Algebraic Geometry
Volume 28
Issue 4
Pages 773-815
ISSN 1056-3911
Publication year 2019
Published at Department of Mathematical Sciences
Pages 773-815
Language en
Keywords Mathematics, ligne p, 1987, inventiones mathematicae, v89, p247, hneider m, 1972, inventiones mathematicae, v16, p161
Subject categories Mathematics


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.

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