Filling in holes of analytic surfaces and classification of compact surfaces (Q1313318)

We consider the problem of filling in holes in dimension 2 where we examine under which condition a strictly pseudoconvex hypersurface in an analytic surface is the boundary of a Stein space. We show that Rossi's example of a strictly pseudoconvex hypersurface \(\Sigma\), which bounds two nonrelatively compact domains, is not the boundary of a Stein space although holomorphic functions in a neighbourhood of \(\Sigma\) give local charts. We show that in a compact complex surface \(S\) without nonconstant meromorphic functions with \(b_ 1(S)=1\), such a phenomenon cannot exist.