Nonexistence of certain Levi flat hypersurfaces in Kähler manifolds from the viewpoint of positive normal bundles
type: Item
instance of: scholarly article
title: Nonexistence of certain Levi flat hypersurfaces in Kähler manifolds from the viewpoint of positive normal bundles
zbMATH Open document ID: 1310.32042
DOI: 10.4171/PRIMS/104
author: Takeo Ohsawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2013-07-24T00:00:00Z
reviewed by: Alberto Saracco
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6190618
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: Levi flat hypersurface
zbMATH Keywords: \(L^2\) estimates for \(\overline\partial\)
MaRDI profile type: Publication
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review text: In this paper, which is dedicated to the memory of Marco Brunella, the author combines the \(L^2\) technique for solving the \(\overline\partial\)-equation with a method by Brunella for proving two nonexistence theorems for Levi flat hypersurfaces.NEWLINENEWLINEIn both theorems, \(X\) is a compact Kähler manifold of dimension at least \(3\) and \(M\) is a Levi-flat hypersurface in \(X\). Then certain positiveness conditions of the curvature are proved to be impossible. Namely:NEWLINENEWLINE1. Suppose \(M\) is of class \(\mathcal C^{2,\alpha}\), \(\alpha>0\), and \(M\) is the union of leaves of a holomorphic foliation \(\mathcal F\) in a neighbourhood \(U\) of \(M\). Then the normal bundle of \(\mathcal F\) does not admit any fiber metric whose curvature form restricted to the tangent space of \(\mathcal F\) is semipositive of rank at least \(2\) at every point of \(M\).NEWLINENEWLINE2. Suppose \(M\) is of class \(\mathcal C^\infty\). Then the holomorphic normal bundle of \(M\) does not admit a fiber metric whose curvature form is semipositive of rank at least \(2\) on the holomorphic tangent space of \(M\).
description: scientific article; zbMATH DE number 6190618