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POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD $\partial \overline {\partial }$-LEMMA

Part of: Global differential geometry Deformations of analytic structures Families, fibrations Commutative algebra: Homological methods

Published online by Cambridge University Press:  08 June 2021

SHENG RAO Open the ORCID record for SHENG RAO [Opens in a new window] ,
XUEYUAN WAN Open the ORCID record for XUEYUAN WAN [Opens in a new window]  and
QUANTING ZHAO Open the ORCID record for QUANTING ZHAO [Opens in a new window]
SHENG RAO
Affiliation:
School of Mathematics and statistics Wuhan UniversityWuhan430072China Department of Mathematics University of California at Los AngelesLos Angeles, California90095-1555USAlikeanyone@whu.edu.cn
XUEYUAN WAN
Affiliation:
Mathematical Science Research Center Chongqing University of TechnologyChongqing400054Chinaxwan@cqut.edu.cn
QUANTING ZHAO
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal UniversityWuhan430079P.R. Chinazhaoquanting@126.com; zhaoquanting@mail.ccnu.edu.cn
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Abstract

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

MSC classification

Primary: 32G05: Deformations of complex structures
Secondary: 13D10: Deformations and infinitesimal methods 14D15: Formal methods; deformations 53C55: Hermitian and Kählerian manifolds
Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 305 - 354
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

*

Rao is partially supported by the National Natural Science Foundations of China No. 11671305, 11771339, 11922115 and China Scholarship Council/University of California, Los Angeles Joint Scholarship Program.

†

Wan is partially supported by Scientific Research Foundation of Chongqing University of Technology.

‡

Zhao is partially supported by the National Natural Science Foundations of China No. 11801205.

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