Compact Stein surfaces with boundary as branched covers of \(B^4\)
A smooth real-valued function \(f :X \rightarrow \mathbb{R}\) on a complex manifold \(X\) is plurisubharmonic iff the complex Hessian \(H f=(\sigma^{2} f/ \sigma z_{i} \sigma {\bar z_{j}})\) is everywhere positively semidefinite for any local complex coordinates \((z_{1},\dots,z_{n})\). A Stein surface is a non-singular complex surface \(X\) which admits a proper strictly plurisubharmonic function \(f:X\rightarrow [0,\infty)\) such that Bd \(X\) is a level set. In this paper the authors prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of \(B^{4}\) whose branch set is a positive braided surface. As a consequence, a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.
scientific article; zbMATH DE number 1580027