Algebraic and analytic cohomology of quasiprojective varieties
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.
comparison of algebraic and analytic cohomology groups
scientific article; zbMATH DE number 4179761
scientific article