On atypical values and local monodromies of meromorphic functions (Q2736073)

Let \(M\) be an \(n\)-dimensional compact complex manifold, and let \(f = P/Q,\) where \(P\) and \(Q\) are nonzero sections of a line bundle over \(M\) so that \(f : M \rightarrow {\mathbf P}^1_{\mathbb C}.\) The authors call \(f\) the meromorphic function on \(M.\) First they remark that the map \(f : M\setminus\{P=Q=0\} \rightarrow {\mathbf P}^1_{\mathbf C}\) is a \(C^\infty\) locally trivial fibration outside a finite subset of the projective line. The smallest such subset \(B(f) \subset {\mathbf P}^1_{\mathbb C}\) is called the bifurcation set of the meromorphic function \(f\) and its elements are called atypical values of \(f.\) Similarly to the case where \(M = {\mathbf P}^n_{\mathbb C}\) [see St. Petersbg. Math. J. 11, No. 5, 775-780 (2000); translation from Algebra Anal. 11, No. 5, 92-99 (2000; Zbl 0972.32022)] the authors introduce the local monodromy operators corresponding to simple loops around the atypical values of \(f\) and the monodromy group of \(f.\) Then the zeta-function \(\zeta_f^c(t)\) of the local monodromy of \(f\) corresponding to \(c\in {\mathbf P}^1_{\mathbb C}\) is defined. For the atypical values \(c\in B(f)\) the zeta-function is expressed via the integral with respect to the Euler characteristic [\textit{O. Viro}, Lect. Notes Math. 1346, 127-138 (1988; Zbl 0686.14019)]. As a consequence the following formula for Euler characteristic of the generic fibre \(F\) is obtained: NEWLINE\[NEWLINE\chi(F) = \chi(M) - \chi(\{Q=0\}) + (-1)^{n-1} \Bigl( \sum\mu_f(c) - \mu_f(\infty)+ \sum\lambda_f(c) -\lambda_f(\infty) \Bigr),NEWLINE\]NEWLINE where \(\mu_f(c)\) and \(\lambda_f(c)\) are generalizations of the Milnor number and the invariant \(\lambda\) studied in [\textit{E. Artal Bartolo, I. Luengo} and \textit{A. Melle-Hernández}, Math. Z. 233, No. 4, 679-696 (2000; Zbl 0956.32025)] for the case of a polynomial \(P\) defined on \({\mathbb C}^n.\)NEWLINENEWLINEFor the entire collection see [Zbl 0967.00102].