Analogs of Lefschetz theorems for linear systems with isolated singularities
Let \({\mathcal D}\) be a base point free linear system on an n-dimensional complex manifold X. The author introduces the notion of a linear system of Lefschetz type and he proves the Lefschetz type theorems for the section of a generic member of \({\mathcal D}\). He shows that this notion behaves well under the pull back \(f^*| {\mathcal D}\), where f: \(Y\to X\) is a cyclic branched covering, branched along a smooth codimension one submanifold \({\mathcal S}\) such that \({\mathcal D}| {\mathcal S}\) is also of Lefschetz type. He uses these results to construct simply connected algebraic surfaces \(Y_ k(x_ 1,...,x_ k)\), \(x_ i\in {\mathbb{N}}\) which give infinitely many homeomorphic but not diffeomorphic algebraic surfaces of general type. The surface \(Y_ k(x_ 1,...,x_ k)\) is the composition of k finite triple cyclic coverings starting from \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) with a smooth branch locus which is linearly equivalent to \(3x_ 1C\) with \(C=pt\times {\mathbb{P}}^ 1+{\mathbb{P}}^ 1\times pt\).
scientific article; zbMATH DE number 4155460