KOHN-ROSSI COHOMOLOGY AND ITS APPLICATION TO THE COMPLEX PLATEAU PROBLEM, II

By Rong Du, S. Yau. 2007

72 citations recorded by the source corpus.

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n − 1 in 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 ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. 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 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/811208e6503f67500375360af8121d1f31eaf1d3