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Kohn-Rossi cohomology and its application to the complex Plateau problem. II. (English) Zbl 1123.32020

Summary: Let \(X\) be a compact connected strongly pseudoconvex CR Manifold of real dimension \(2n-1\) in \(\mathbb C^{n+1}\). Tanaka introduced a spectral 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 \(H^k_h(X)\). We study the holomorphic de Rham cohomology in terms of the \(s\)-invariant of the isolated singularities of the variety \(V\) bounded by \(X\). We give a characterization of the singularities with vanishing \(s\)-invariants. For \(n\geq 3\), S. S.-T. Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981 [part I of this paper, Ann. Math. (2) 113, 67–110 (1981; Zbl 0464.32012)]. For \(n = 2\), the problem has remained 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 \(\dim X = 3\) and \(H^2_h(X)= 0\), then \(X\) is a boundary of a 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.

MSC:

32V40 Real submanifolds in complex manifolds
32S05 Local complex singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32C36 Local cohomology of analytic spaces

Citations:

Zbl 0464.32012
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