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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.

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 |