Kohn-rossi cohomology and its application to the complex plateau problem, ii

Hing Sun Luk The Chinese University of Hong Kong Stephen S.-T. Yau East China Normal University

Differential Geometry mathscidoc:1609.10033

Journal of Differential Geometry, 77, (1), 135-148, 2007
Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n.1 in Cn+1. Tanaka introduced a spec- tral sequence E(p,q) r (X) with E(p,q) 1 (X) being the Kohn-Rossi cohomology group and E(k,0) 2 (X) being the holomorphic De Rham cohomology denoted by Hk h(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated sin- gularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has re- mained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dimX = 3 and H2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.
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@inproceedings{hing2007kohn-rossi,
  title={KOHN-ROSSI COHOMOLOGY AND ITS APPLICATION TO THE COMPLEX PLATEAU PROBLEM, II},
  author={Hing Sun Luk, and Stephen S.-T. Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908203819143208690},
  booktitle={Journal of Differential Geometry},
  volume={77},
  number={1},
  pages={135-148},
  year={2007},
}
Hing Sun Luk, and Stephen S.-T. Yau. KOHN-ROSSI COHOMOLOGY AND ITS APPLICATION TO THE COMPLEX PLATEAU PROBLEM, II. 2007. Vol. 77. In Journal of Differential Geometry. pp.135-148. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908203819143208690.
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