On the plurisubharmonicity of the leafwise Poincaré metric on projective manifolds
type: Item
instance of: scholarly article
title: On the plurisubharmonicity of the leafwise Poincaré metric on projective manifolds
zbMATH Open document ID: 1101.32015
author: Marco Brunella
published in: Journal of Mathematics of Kyoto University
publication date: 2006-03-23T00:00:00Z
review text: A \textit{skew cylinder} (in Yu. S. Ilyashenko's terminology, see [\textit{Yu. S. Il'yashenko} and \textit{A. A. Shcherbakov}, Tr. Mat. Inst. Steklova. 213, 112--122 (1997; Zbl 0911.32033)]) is a complex manifold holomorphically fibered over another one by simply connected holomorphic curves that admits a holomorphic section. The paper under review studies an important class of skew cylinders that are related to foliations with singularities (of complex codimension at least 2) by analytic curves on complex projective manifolds. To each (topologically trivial) cross-section \(S\) of a foliation one associates the \textit{universal covering manifold} \(U_S\) [see \textit{Y. S. Ilyashenko}, Topol. Methods Nonlinear Anal. 11, No.~2, 361--373 (1998); remark ibid. 23, No.~2, 377--381 (2004; Zbl 0927.32020), Math. USSR, Sb. 17(1972), 551--569 (1973); translation from Mat. Sb., N. Ser. 88(130), 558--577 (1972; Zbl 0243.57011)]. This is a union of a family of simply connected Riemann surfaces (with marked points) that is parametrized by points of \(S\): for each \(s\in S\) the corresponding surface is the universal covering over the leaf passing through \(s\) with the marked point \(s\). It is known (see [\textit{M. Brunella}, On entire curves tangent to a foliation, preprint (2004)]) that in almost all the cases the space \(U_S\) admits a natural structure of complex manifold that is a skew cylinder fibered over \(S\). A similar result in the case, when the foliated manifold is Stein was proved by Yu. S. Ilyashenko (see his two latter loc. cit.), who had also shown that in this case the universal covering manifold is Stein. There is an important open question: is it true that the universal covering manifold \(U_S\) is always \textit{Stein} (whenever it is a manifold and the foliated manifold is compact projective)? The paper under review provides a proof of a metric version of this statement. It is known [\textit{H. Yamaguchi}, Jap. J. Math. 7, 319--377 (1981; Zbl 0485.3201…
reviewed by: Alexey A. Glutsyuk
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 5014811
zbMATH Keywords: holomorphic foliation
zbMATH Keywords: Riemann surface
zbMATH Keywords: Poincaré metric
zbMATH Keywords: Bergman metric
zbMATH Keywords: Stein manifold
zbMATH Keywords: plurisubharmonic function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250281996
OpenAlex ID: W1592888393
DOI: 10.1215/KJM/1250281996
Recommended article: Covering manifolds for analytic families of leaves of foliations by analytic curves
Recommended article: The exhaustion method for skew cylinders
Recommended article: Almost complex structures on universal coverings of foliations
Recommended article: Uniformization of foliations with hyperbolic leaves and the Beltrami equation with parameters
Recommended article: Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem
Recommended article: Q4884388
Recommended article: Closed holomorphic 1-forms without zeros on Stein manifolds
Recommended article: Uniformization and the Poincar� metric on the leaves of a foliation by curves
Recommended article: Uniformisation of foliations by curves
Recommended article: Q3762608
description: scientific article; zbMATH DE number 5014811