Meromorphic functions, bifurcation sets and fibred links (Q2459327)

Suppose that \(f,g: ({\mathbb C}^n,0)\to({\mathbb C},0)\) are germs of holomorphic functions without common factors such that \(f/g\) is semitame: that is, the bifurcation set \(B\) consisting of all limit points of \(f(x)/g(x)\) as \(x\to 0\) is \(\{0,\infty\}\). Let \(L_f\subset {\mathbb S}_\varepsilon^{2n-1}\) be the link of \(f\) in a sphere of radius \(\varepsilon\) centred at the origin. Then the Milnor map \[ {{f/g}\over{| f/g| }}: {\mathbb S}_\varepsilon^{2n-1}\setminus (L_f\cup L_g) \longrightarrow {\mathbb S}^1 \] is a \(C^\infty\) locally trivial fibration. For \(n=2\) it is a fibration of the link \(L_{f/g}=L_f\cup -L_g\). This is proved here by a relatively minor modification of a standard argument of Milnor. More striking is the authors' converse result for \(n=2\). If \(f,g: ({\mathbb C}^2,0)\to({\mathbb C},0)\) are holomorphic germs with no common branch, then the following conditions are shown to be equivalent: (1) \(L_{f/g}\) is a fibred link; (2) the real analytic germ \(f\bar{g}\) has an isolated critical value at~\(0\); (3) the Milnor map is a \(C^\infty\) local fibration; (4) \(f/g\) is semitame at the origin; (5) every \(c\neq 0,\infty\) is a generic value of the local pencil generated by \(f\) and \(g\). This is also applied to global maps \(f,g:{\mathbb C}^2\to{\mathbb C}\). The authors give a semitameness condition at \(\infty\) for \(f/g\) and show that it is satisfied if and only if the link at infinity \(L_{f/g,\infty}=L_{f,\infty}\cup -L_{g,\infty}\) is fibred (always by the Milnor map).