Théorème de dualité pour la cohomologie locale au bord des ouverts fortement pseudoconvexes. (Duality theorem for local cohomology at the boundary of strongly pseudoconvex open sets)
Let D be a relatively compact open set of a complex manifold M of dimension n (n\(\geq 2)\) such that the boundary \(\partial D=B\) is a smooth strongly pseudoconvex \({\mathcal C}^{\infty}\) manifold. For \({\mathcal F}^ a \)coherent analytic sheaf on M the author proves the following results: \((1)\quad \dim_{{\mathbb{C}}}H^ q(B;\underline H^ 1_ B({\mathcal F}))<\infty\) for \(0<q<prof {\mathcal F}-1;\) (2) the spaces \(H^ q(B;\underline H^ 1_ B({\mathcal F}))\) and \(Ext^{n-1-q}(B;\underline H^ 1_ B({\mathcal F}),\underline H^ 1_ B(\Omega^ n))\) are dual topological vector spaces for \(q<prof {\mathcal F}-1.\) These results are the analogues for the sheaf \b{H}\({}^ 1_ B({\mathcal F})\) of the results of A. Andreotti and H. Grauert for (1) and of B. Malgrange and J.-P. Serre for (2). The techniques are standard.
scientific article; zbMATH DE number 3939699