A generalization of Lefschetz theorem (Q580455)

Using Morse theory, the author proves the following improvement of Lefschetz theorem: let V be a complex n-dimensional algebraic variety, A an ample effective divisor on it and \(x\in V\setminus A\) a point such that \(V\setminus \{A\cup \{x\})\) is smooth; then \(\pi_ k(V\setminus \{x\},A)=1\) for \(k<n\). Next this result is applied in the situation when x is the vertex of the cone obtained by contracting the zero section of an ample line bundle L on a compact manifold M, thus obtaining informations about the maps \(\pi_ k(X)\to \pi_ k(M)\) where X is any compact analytic subspace of L of pure dimension equal to that of M.