Kohn-Rossi cohomology and the complex Plateau problem (Q2671286)

The authors study the so-called complex Plateau problem: which odd dimensional real submanifolds \(X\) of \(\mathbb C^N\) are boundaries of Stein manifolds [\textit{F. R. Harvey} and \textit{H. B. Lawson jun.}, Ann. Math. (2) 102, 223--290 (1975; Zbl 0317.32017)]. If \(V\) has hypersurface singularities only, \textit{S. S. T. Yau} gave an answer to this problem by calculations of the Kohn-Rossi cohomology groups \(H^{p,q}(\partial V)\) [Ann. Math. (2) 113, 67--110 (1981; Zbl 0464.32012)]. In the paper under review the authors describe similar results when \(V\) has at most isolated complete intersection singularities. Their main idea is to represent the dimensions of \(H^{p,q}(X, \mathcal F)\) as a sum of \(\dim_{\mathbb C} H_{\{x\}}^{q+1}(X, \Omega^p\otimes\mathcal F)\), where \(X\) is the smooth boundary of an \(n\)-dimensional reduced irreducible Stein space \(V\), \(\mathcal F\) is a coherent analytic sheaf locally free in a suitable neighborhood \(V'\) of \(V\) and \(\Omega^p\) are sheaves of holomorphic \(p\)-forms on \(V'\).