Abstract
Let X 1 and X 2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions 2m - 1 and 2n - 1 in ℂm+1 and ℂn+1, respectively. We introduce the Thom-Sebastiani sum X = X 1⊕X 2 which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in ℂm+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in ℂn+1 for all n ≥ 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X 1⊕X 2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X 1 ⊕ X 2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X 1 and X 2 provided that X 2 admits a transversal holomorphic S 1-action.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11531007 and 11401335), Start-Up Fund from Tsinghua University and Tsinghua University Initiative Scientific Research Program. The first author thanks National Center for Theoretical Sciences for providing excellent research environment while part of this research was done.
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In memory of Professor LU QiKeng (1927–2015)
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Yau, S.S.T., Zuo, H. Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds. Sci. China Math. 60, 1129–1136 (2017). https://doi.org/10.1007/s11425-016-5125-6
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DOI: https://doi.org/10.1007/s11425-016-5125-6