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Deformations of Dolbeault cohomology classes

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Abstract

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer–Cartan equation. Following the classical theory of Kodaira–Spencer–Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of (p, q)-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura’s example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology.

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Notes

  1. Though the base space B may be a singular complex space, such as the Kuranishi space, only deformations of complex manifolds will be considered. Thus every fiber of \(\pi \) are complex manifolds. This is our basic setting in this paper.

  2. It turns out that the effect of the former is equivalent to the obstructions of classes in \(H^{0,q-1}_{{\bar{\partial }}}(X,E)\), see Proposition 4.6.

  3. Even for canonical deformations, it is not clear how they depend on the choices of the Hermitian metric. For example, is it true that the equivalence class of canonical deformations of a given class is unique (thus independent of the Hermitian metrics chosen)?

  4. Here and throughout this paper, G will always denote the \({\bar{\partial }}\)-Green operator unless otherwise stated.

  5. To save space, we have omitted \(\cap A^{0,q}(X,E)\) which will be clear from the context.

  6. In fact, a more general result of this type holds, see [22, pp. 210].

  7. We have been informed by the anonymous referee that by using the structure theory of double complexes one can show \(\partial _{A,{\bar{\partial }}(\ker \partial )}^{p,q}=0\) holds for all \((p,q)\in {\mathbb {Z}}^2\) is equivalent to the \(\partial {\bar{\partial }}\)-lemma on X, see [30, 49, 52, 63] for more information.

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Acknowledgements

I would like to thank Prof. Kefeng Liu for his constant encouragement and many useful discussions. Many thanks to Sheng Rao, Quanting Zhao, Guillaume Rond, Xiaokui Yang, Kang Wei, Yang Shen, Shengmao Zhu, Kai Tang, Chunle Huang, Kai Liu and Ruosen Xiong for useful communications. I would also like to thank Prof. Bing-Long Chen for his constant support. I am very grateful to the anonymous referees for their careful reading and for many helpful suggestions.

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  1. Mathematical Science Research Center, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China

    Wei Xia

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Correspondence to Wei Xia.

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This work was supported by the National Natural Science Foundation of China No. 11901590 and Scientific Research Foundation of Chongqing University of Technology.

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Xia, W. Deformations of Dolbeault cohomology classes. Math. Z. 300, 2931–2973 (2022). https://doi.org/10.1007/s00209-021-02900-w

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