Analogs of Lefschetz theorems for linear systems with isolated singularities
type: Item
instance of: scholarly article
title: Analogs of Lefschetz theorems for linear systems with isolated singularities
zbMATH Open document ID: 0704.57009
published in: Journal of Differential Geometry
publication date: 1990-01-01T00:00:00Z
review text: 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 \(C=pt\times {\mathbb{P}}^ 1+{\mathbb{P}}^ 1\times pt\).
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 14F35
zbMATH DE Number: 4155460
zbMATH Keywords: linear system on complex manifold
zbMATH Keywords: linear system of Lefschetz type
zbMATH Keywords: cyclic branched covering
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: homeomorphic but not diffeomorphic
Wikidata QID: Q115182409
author: Boris Moishezon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214444089
OpenAlex ID: W1568601854
DOI: 10.4310/JDG/1214444089
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description: scientific article; zbMATH DE number 4155460