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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 2984–3047 | Cite as

Several Special Complex Structures and Their Deformation Properties

  • Sheng Rao
  • Quanting ZhaoEmail author
Article

Abstract

We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang, and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the Kähler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a Kählerian family is contained in the closure of the Gauduchon cone for this fiber.

Keywords

Deformations of complex structures Deformations and infinitesimal methods Formal methods Deformations Hermitian and Kählerian manifolds 

Mathematics Subject Classification

Primary 32G05 Secondary 13D10 14D15 53C55 

Notes

Acknowledgements

We would like to express our gratitude to Professors Daniele Angella, Kwokwai Chan, Huitao Feng, Jixiang Fu, Lei Fu, Conan Leung, Kefeng Liu, Dan Popovici, Fangyang Zheng, and Dr. Jie Tu, Yat-hin Suen, Xueyuan Wan, Jian Xiao, Xiaokui Yang, Wanke Yin, Shengmao Zhu for their useful advice or interest on this work. This work started when the first author was invited by Professor J.A. Chen to Taiwan University in May–July of 2013 with the support of the National Center for Theoretical Sciences, and was completed during his visit in June of 2015 to the Mathematics Department of UCLA. He takes this opportunity to thank them for their hospitality. Last but not least, the anonymous referee’s careful reading and valuable comments improve the statement significantly. Rao is partially supported by the National Natural Science Foundations of China No. 11301477, 11671305, and China Scholarship Council/University of California, Los Angeles Joint Scholarship Program. The corresponding author Zhao is partially supported by the Fundamental Research Funds for the Central Universities No. CCNU16A05013 and China Postdoctoral Science Foundation No. 2016M592356

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Copyright information

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  3. 3.School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanPeople’s Republic of China

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