Der Kodairasche Verschwindungssatz auf streng pseudokonvexen Räumen. II
type: Item
instance of: scholarly article
title: Der Kodairasche Verschwindungssatz auf streng pseudokonvexen Räumen. II
zbMATH Open document ID: 0538.32023
DOI: 10.1007/BF01455307
author: Thomas Peternell
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182983
review text: This is the second of two papers [for part I see the review above Zbl 0538.32022] in which the Kodaira vanishing theorem on an irreducible reduced strongly pseudoconvex complex space X is proved: \(H^ q(X,{\mathcal E}\otimes {\mathcal K}_ X)=0\) for \(q>0\) and any semi-positive locally free sheaf \({\mathcal E}\) on X with a suitable canonical sheaf \({\mathcal K}_ X\). The first part dealt with a special - namely the algebraic - case. Now in this second part the theorem in the general non-algebraic case is proved using essentially the solution of the special case. The main tool is the method of algebraic approximation. If X is a normal strongly pseudoconvex space with exceptional set A and ideal sheaf \({\mathcal J}\) defining A in X, then an algebraic strongly pseudoconvex X' (i.e. X' has a Moishezon compactification) with exceptional set A' and ideal sheaf \({\mathcal J}'\) defining A' is called an algebraic approximation of order \(\mu\in {\mathbb{N}}\) if \((A,0_ X/{\mathcal J}^{\mu +1})\simeq(A',\quad 0_{X'}/{\mathcal J}^{\prime\mu +1})\). Then the proof of the existence of algebraic approximations of arbitrary high order and with sufficiently nice properties constitutes the technical heart of the paper. - At the end of the paper some generalizations for coherent sheaves are proved.
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 3855496
zbMATH Keywords: Kodaira vanishing theorem
zbMATH Keywords: irreducible reduced strongly pseudoconvex complex space
zbMATH Keywords: algebraic approximation
MaRDI profile type: Publication
cites work: Théorèmes de finitude pour la cohomologie des espaces complexes
cites work: Algebraic approximation of structures over complete local rings
cites work: On the solutions of analytic equations
cites work: Complex analytic geometry
cites work: \(q\)-konvexe Restmengen in kompakten komplexen Mannigfaltigkeiten
cites work: On the blowing down of analytic spaces
cites work: Über Modifikationen und exzeptionelle analytische Mengen
cites work: Q5638554
cites work: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen
cites work: Q5611966
cites work: Q4195061
cites work: Q3125202
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Flattening Theorem in Complex-Analytic Geometry
cites work: On the equivalence of imbeddings of exceptional complex spaces
cites work: Relativexzeptionelle analytische Mengen. (Relative exceptional analytical sets.)
cites work: On a Differential-Geometric Method in the Theory of Analytic Stacks
cites work: Harmonic integrals on strongly pseudo-convex manifolds. I
cites work: On complex analytic vector bundles
cites work: Vektorraumbündel in der Nähe von kompakten komplexen Unterräumen
cites work: On strongly pseudo-convex Kaehler manifolds
cites work: Der Kodairasche Verschwindungssatz auf streng pseudokonvexen Räumen. I
cites work: Characterizing Moišezon spaces by almost positive coherent analytic sheaves
cites work: Q5570369
cites work: Familien negativer Vektorraumbündel und 1-konvexe Abbildungen
cites work: A vanishing theorem for semipositive line bundles over non-Kähler manifolds
cites work: Gap-sheaves and extension of coherent analytic subsheaves
description: scientific article; zbMATH DE number 3855496