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Complex Manifolds

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A differential-geometric approach to deformations of pairs (X, E)

Kwokwai Chan
  • Corresponding author
  • Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yat-Hin Suen
  • Corresponding author
  • Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-02-01 | DOI: https://doi.org/10.1515/coma-2016-0002


This article gives an exposition of the deformation theory for pairs (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, adapting an analytic viewpoint `a la Kodaira- Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer–Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of E, obtaining a chain level refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of E over X. As an application, we give examples where deformations of pairs are unobstructed.


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About the article

Received: 2015-09-02

Accepted: 2016-01-21

Published Online: 2016-02-01

Citation Information: Complex Manifolds, Volume 3, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2016-0002.

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© 2016 Kwokwai Chan and Yat-Hin Suen. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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