Fundamental groups of open algebraic varieties
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 surjective with kernel an abelian group generated by at most \(m\) elements contained in the centre of \(\pi_1 (U)\). Further, if \(p\) is also general, then this representation is an isomorphism. The second application is the determination of the fundamental group of the complement of a projective plane curve \(f^q+ g^p =0\), \(p\geq 2\) and \(q\geq 2\) being degrees of the two general homogeneous polynomials \(f\) and \(g\). Such results are due to \textit{O. Zariski} [Am. J. Math. 51, 305-328 (1929)], \textit{M. Oka} [Math. Ann. 218, 55-65 (1975; Zbl 0335.14005) and J. Math. Soc. Japan 30, 579-597 (1978; Zbl 0387.14004)]\ and \textit{A. Nemethi} [Math. Proc. Camb. Philos. Soc. 102, 453-457 (1987; Zbl 0679.14004)].
topological fundamental groups of complex algebraic varieties
fundamental group of the complement of a projective plane curve
scientific article; zbMATH DE number 794437