Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds |
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Authors: | Stephen S T Yau HuaiQing Zuo |
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Institution: | 1.Department of Mathematical Sciences,Tsinghua University,Beijing,China;2.Yau Mathematical Sciences Center,Tsinghua University,Beijing,China |
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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 ? m+1 and ? n+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 ? m+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in ? n+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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