Topology of the complement of a projective algebraic set
The author uses the method of hyperplane sections introduced by S. Lefschetz to study the complement \(P^ n(C)\setminus A\) of a closed algebraic system \(A\) in a complex projective space. He proves the following theorem: Let \(A\) be a closed algebraic system of codimension \(q\) in the complex projective space \(P^ n(C)\) with \(n\geq 1\). Let \(S\) be a Whitney stratification of \(A\) and let \(L\) be a projective hyperplane of \(P^ n(C)\) transversal to all strata of \(S\). Under these conditions the canonical injection \(L\cap(P^ n(C)\setminus A)\hookrightarrow P^ n(C)\setminus A\) induces isomorphisms of the \(k\)th singular homology groups with integer coefficients if \(k\leq n+q-3\) and a surjective homomorphism if \(k=n+q-2\). The choice of \(L\) is generic for this statement. As a corollary one gets a respective theorem about the homotopy groups via a result of Lefschetz. The extensive proof refines the classical methods with regard to the stratification. The homology of the fibration and the first isotopy theorem of Thom-Mather are used in the most important parts of the proof.
scientific article; zbMATH DE number 55795