Parabolic exhaustions and analytic coverings
type: Item
instance of: scholarly article
title: Parabolic exhaustions and analytic coverings
zbMATH Open document ID: 0791.32013
DOI: 10.1007/BF02571722
author: Finnur Lárusson
published in: Mathematische Zeitschrift
publication date: 1994-07-11T00:00:00Z
full work available at URL: https://eudml.org/doc/174620
review text: Let \(\tau\) be a parabolic exhaustion on a Stein manifold \(X\) such that \(\tau\) is strictly plurisubharmonic at its zeros. The metric defined by \(\tau\) on the complement of its degeneracy locus \(D\) is shown to be flat if \(\tau\) is real-analytic or if ``most'' leaves of the associated Monge- Ampère foliation \({\mathcal F}\) abut the zeros of \(\tau\). Then, by an analysis of the singularities of \(\tau\), we show that the tangent bundle of \(X \backslash D\) extends to a flat hermitian bundle on \(X\) with a holomorphic section \(s\) such that \(\tau=\| s |^ 2\), and that \({\mathcal F}\) extends to a singular holomorphic foliation of \(X\). Also, \(\tau\) is the length-squared of an analytic covering of \(X\) onto a ball if and only if the monodromy of the \(\tau\)-connection is trivial. We obtain a characterization of affine algebraic manifolds as those \(X\) possessing \(\tau\) with finite monodromy and affine leaves.
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 567410
zbMATH Keywords: analytic coverings
zbMATH Keywords: Monge-Ampère foliations
zbMATH Keywords: parabolic exhaustion
zbMATH Keywords: Stein manifold
zbMATH Keywords: affine algebraic manifolds
MaRDI profile type: Publication
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OpenAlex ID: W2063607616
description: scientific article; zbMATH DE number 567410