1-convex manifolds are \(p\)-Kähler
type: Item
instance of: scholarly article
title: 1-convex manifolds are \(p\)-Kähler
zbMATH Open document ID: 1023.32010
DOI: 10.1007/BF02941676
author: M. Leoni
author: Giovanni Bassanelli
author: Lucia Alessandrini
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2003-02-13T00:00:00Z
review text: A complex manifold \(X\) is called \(p\)-Kähler if it carries a closed transverse (i.e. strictly weakly positive) \((p,p)\)-form \(\omega\). \(\omega\) is called a \(p\)-Kähler form. Kähler manifolds are 1-Kähler and also \(p\)-Kähler for each \(p\) such that \(1 \leq p \leq\dim X\), therefore this notion is relevant in the non-Kähler case. The authors prove that if \(X\) is a 1-convex manifold possessing an exceptional set \(S\) with \(\dim S=k\), then \(X\) is \(p\)-Kähler for every \(p\) greater than \(k\). Also, \(X\) is \(k\)-Kähler iff the ``fundamental class'' of \(S\) does not vanish. Several further interesting discussions and examples are also presented.
Mathematics Subject Classification ID: 32F10
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32C35
zbMATH DE Number: 1868559
zbMATH Keywords: 1-convex manifold
zbMATH Keywords: Kähler manifold
zbMATH Keywords: strictly weakly positive \((p,p)\)-form
zbMATH Keywords: positive current
reviewed by: Eugen Pascu
MaRDI profile type: Publication
cites work: Q3756095
cites work: Compact p-Kähler manifolds
cites work: Positive \(\partial\bar\partial\)-closed currents and non-Kähler geometry
cites work: Metric properties of manifolds bimeromorphic to compact Kähler spaces
cites work: Compact complex threefolds which are Kähler outside a smooth rational curve
cites work: On the embedding of 1-convex manifolds with 1-dimensional exceptional set
cites work: Théorèmes de finitude pour la cohomologie des espaces complexes
cites work: A cut-off theorem for plurisubharmonic currents
cites work: Q3872748
cites work: Convexité de l'espace des cycles
cites work: Q5601005
cites work: Q5628487
cites work: Q5640205
cites work: Q4744546
cites work: Q4352973
cites work: Kählerian normal complex surfaces
cites work: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen
cites work: An intrinsic characterization of Kähler manifolds
cites work: Q4775126
cites work: On the existence of special metrics in complex geometry
cites work: Extension theorems for Kähler metrics
cites work: On 𝑛-dimensional compact complex varieties with 𝑛 algebraically independent meromorphic functions
cites work: The Levi problem for complex spaces. II
cites work: On the homology groups of Stein spaces
cites work: Remarques sur la cohomologie des variétés analytiques complexes
cites work: Holomorphic convexity of spaces of analytic cycles
cites work: On strongly pseudo-convex Kaehler manifolds
cites work: On certain non-Kählerian strongly pseudoconvex manifolds
description: scientific article; zbMATH DE number 1868559