Fundamental groups of open algebraic varieties
type: Item
instance of: scholarly article
title: Fundamental groups of open algebraic varieties
zbMATH Open document ID: 0840.14010
DOI: 10.1016/0040-9383(94)00045-M
author: Ichiro Shimada
published in: Topology
publication date: 1996-07-04T00:00:00Z
review text: The author gives a method of computing topological fundamental groups of complex algebraic varieties by expressing them as fibres of good fibrations and relating their fundamental groups to those of the total space of the fibration. Let \(Y\) and \(B\) be nonsingular algebraic varieties over \(\mathbb{C}\). Let \(p: Y\to B\) be a dominant morphism with general fibre \(Y_0\) connected. Assume that \(p\) has a section \(s\) over a classical open set \(U\subset B\). Under certain conditions on \(s\) and \(U\), the author determines the kernel of the morphism \(\pi_1 (Y_0, s(0))\to \pi_1 (p^{-1} (U), s(0))\). He also gives a set of sufficient conditions for this kernel to be trivial. This is used to prove the main theorem: Let \(F\) be a nonsingular connected quasi-projective variety of dimension \(\geq 2\) and \(\overline {F}\) its nonsingular projective compactification such that \(\overline {F}- F\) is of codimension \(\geq 2\) in \(\overline {F}\). Let \(W\subset F\times \mathbb{A}^n\) be a reduced (possibly reducible) effective divisor, \(W_a=\) fibre over \(a\in \mathbb{A}^n\). Assume that the locus of \(a\in \mathbb{A}^n\) such that \(W_a\) is not a reduced divisor in \(F\) is of codimension \(\geq 2\) in \(\mathbb{A}^n\). Then the natural homomorphism \(\pi_1 (F-W_a)\to \pi_1 ((F\times \mathbb{A}^n- W)\) is an isomorphism for a general \(a\in \mathbb{A}^n\). Two applications of the theorem are given. For the first, consider general enough hyperplanes \(V_i\subset \mathbb{P}^n\), degree \(V_i= d_i\), \(i=1, \dots, m\). Let \(U\) be the subset of the Grassmannian of lines in \(\mathbb{P}^n\) consisting of all lines which intersect \(V_1\cup \dots \cup V_m\) in distinct \(d_1+ \cdots+ d_m\) points and \(U_p\) the subset of \(U\) consisting of lines passing through a fixed point \(p\in \mathbb{P}^n\). The author shows that for \(n\geq 3\) and \(\sum d_i\geq 3\), the natural monodromy representation of \(\pi_1 (U)\) in the generalized mapping class group is surjectiv…
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 794437
zbMATH Keywords: Zariski problem
zbMATH Keywords: topological fundamental groups of complex algebraic varieties
zbMATH Keywords: good fibrations
zbMATH Keywords: mapping class group
zbMATH Keywords: fundamental group of the complement of a projective plane curve
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 794437