q-completeness of subsets in complex projective space
Let A be a q-codimensional algebraic submanifold of complex projective space. It is proved that \({\mathbb{P}}_ n-A\) is (2q-1)-complete. This implies a vanishing theorem for analytic cohomology groups and the Lefschetz theorem of Larsen on the homotopy of the pair \(({\mathbb{P}}_ n,A).\) Furthermore it is proved that on each open relatively compact subset of a manifold \({\mathbb{P}}_ n-B\) there exists a q-convex function, where B is an algebraic subset of \({\mathbb{P}}_ n\) with \(\dim B=n-q\) and arbitrary singularities. The theorem then follows from the q-convexity of \({\mathbb{P}}_ n-A\) for smooth A.
algebraic submanifold of complex projective space
scientific article; zbMATH DE number 3987661
scientific article