Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Geometry of logarithmic forms and deformations of complex structures


Authors: Kefeng Liu, Sheng Rao and Xueyuan Wan
Journal: J. Algebraic Geom. 28 (2019), 773-815
DOI: https://doi.org/10.1090/jag/723
Published electronically: July 19, 2019
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Abstract | References | Additional Information

Abstract: 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$-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)$-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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Kefeng Liu
Affiliation: Mathematical Sciences Research Center, Chongqing University of Technology, Chongqing 400054, China — and — Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095
Email: liu@math.ucla.edu

Sheng Rao
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Email: raoshengmath@gmail.com, likeanyone@whu.edu.cn

Xueyuan Wan
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, 412 96 Gothenburg, Sweden
Email: xwan@chalmers.se

DOI: https://doi.org/10.1090/jag/723
Received by editor(s): November 20, 2017
Received by editor(s) in revised form: April 3, 2018
Published electronically: July 19, 2019
Additional Notes: The first author was partially supported by NSF (Grant No. 1510216). The second author was partially supported by NSFC (Grants No. 11671305, 11771339).
Article copyright: © Copyright 2019 University Press, Inc.