Algebraic and analytic cohomology of quasiprojective varieties
type: Item
instance of: scholarly article
title: Algebraic and analytic cohomology of quasiprojective varieties
zbMATH Open document ID: 0716.32008
DOI: 10.1007/BF01453585
author: Mathias Peternell
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164647
review text: The main result of the paper is the following: let X be a Zariski open subset of \({\mathbb{P}}^ n_{{\mathbb{C}}}\) and \({\mathcal F}\) a coherent sheaf on X. Then the image of the comparison map \[ \alpha: H^{n-2}(X,{\mathcal F}) \to H^{n-2}(X^{an},{\mathcal F}^{an}) \] is dense. For the proof, the essential ingredient is the following geometric construction: let \(A\subset {\mathbb{P}}^ n_{{\mathbb{C}}}\) be an algebraic subset of pure codimension q. Then there exists a finite surjective map \(\pi: X\to {\mathbb{P}}^ n_{{\mathbb{C}}}\), such that \(\pi^{- 1}(A)=\cup^{r}_{i=1}A_ i\) and \(X\setminus A_ i\) is the union of q open affine subsets. Moreover, a Mayer-Vietoris and a trace argument are used to obtain the desired result for \(X={\mathbb{P}}^ n\setminus A.\) It seems to the reviewer that the density of \(\alpha\) is related to the injectivity of the comparison map \[ H^ 1(A,{\mathcal G}| A) \to H^ 1(A,{\mathcal G}| A^{\bigwedge}) \] (assume \({\mathcal F}\) to be locally free on \({\mathbb{P}}^ n\) and \({\mathcal G}:=\omega_{{\mathbb{P}}^ n}\otimes {\mathcal F}^ v)\) which is in accordance with conjecture (2.3) of the paper of the reviewer and \textit{T. Peternell} [Comp. Math. 74, 299-325 (1990; Zbl 0709.32009)]. From this point of view, some questions posed by the author admit an answer and moreover, in the reviewer's opinion, a generalization of the main result should be possible in this way.
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 32F10
Mathematics Subject Classification ID: 14F99
zbMATH DE Number: 4179761
zbMATH Keywords: comparison of algebraic and analytic cohomology groups
zbMATH Keywords: q-convexity
reviewed by: Siegmund Kosarew
MaRDI profile type: Publication
cites work: Q5563439
cites work: Über die analytische Cohomologiegruppe \(H^{n-1}(\mathcal P_{n} \setminus A,\mathcal J)\)
cites work: Q4159604
cites work: Q4143433
cites work: Cohomological dimension of algebraic varieties
cites work: Formal functions and formal embeddings
cites work: A comparison theorem
cites work: Q5667036
cites work: Über eine Vermutung von Hartshorne
Recommended article: Analytic cohomology groups in top degrees of Zariski open sets in \({\mathbb{P}^n}\)
Recommended article: A comparison theorem
Recommended article: Some Algebraic Sets of High Local Cohomological Dimension in Projective Space
Recommended article: On Barth’s conjecture concerining
Recommended article: Q4724050
Recommended article: A Criterion for the Splitting of a Vector Bundle
Recommended article: Q3035421
Recommended article: Sectional genus and cohomology of projective varieties
Recommended article: Kodaira vanishing theorems on non-complete algebraic manifolds
Recommended article: A Kleiman–Bertini theorem for sheaf tensor products
description: scientific article; zbMATH DE number 4179761