Compact Stein surfaces with boundary as branched covers of \(B^4\)
type: Item
instance of: scholarly article
title: Compact Stein surfaces with boundary as branched covers of \(B^4\)
zbMATH Open document ID: 0983.32027
author: Andrea Loi
author: Riccardo Piergallini
published in: Inventiones Mathematicae
publication date: 2002-04-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0002042
review text: 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.
reviewed by: Claus Ernst
Mathematics Subject Classification ID: 32Q28
Mathematics Subject Classification ID: 32C18
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1580027
zbMATH Keywords: Stein surfaces
zbMATH Keywords: branched covering
zbMATH Keywords: Stein fillable 3-manifold
MaRDI profile type: Publication
OpenAlex ID: W3105046668
arXiv ID: math/0002042
DOI: 10.1007/S002220000106
description: scientific article; zbMATH DE number 1580027