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Dolbeault-type complexes on \(G_2\)- and \(\mathrm {Spin}(7)\)-manifolds

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Abstract

There are three types of Dolbeault complexes arising from representations of holonomy group on a Riemannian manifold, two of which are dual to each other. Such a complex is elliptic if and only if its generator satisfies an algebraic condition. We list all Dolbeault complexes on compact \(G_2\)- and \(\mathrm {Spin}(7)\)-manifolds. Each cohomology group can be described by harmonic forms.

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Acknowledgements

The author would like to thank D.D. Joyce for explanations about the dirac operator on \(\mathrm {Spin}(7)\)-manifold. The author is grateful to Prof. J. Zhou for many useful suggestions. The author is also indebted to H. J. Fan for the guidance over the past years.

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  1. School of Mathematics and Physics, North China Electric Power University, Beijing, China

    Xue Zhang

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  1. Xue Zhang
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Zhang, X. Dolbeault-type complexes on \(G_2\)- and \(\mathrm {Spin}(7)\)-manifolds. manuscripta math. 172, 685–703 (2023). https://doi.org/10.1007/s00229-022-01433-8

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