On the topology of germs of meromorphic functions and its applications

Abstract

Let f = P/Q be a meromorphic function germ on (Cn+1, 0), where P,Q: (Cn+1, 0) ! (C, 0) are holomorphic germs. The authors [Funktsional. Anal. i Prilozhen. 32 (1998), no. 2, 26–35, 95; MR1647824 (99j:32040)] introduced a notion of Milnor fibers and monodromy operators of the germ f around zero and infinity and discussed their properties in connection with the problem of computing the zeta-function of the monodromy. The paper under review is devoted to a further investigation of the basic properties of topological invariants of f. In particular, in the case when P has an isolated singularity at the origin the authors prove that the Euler characteristic of the Milnor 0-fiber of f is equal to (−1)n(μ(P, 0)−μ(P +tQ, 0)) for general t 2 C, where μ(P, 0) is the Milnor number of P at zero. They also describe a generalization of a formula from [A. Parusiński and P. Pragacz, J. Algebraic Geom. 4 (1995), no. 2, 337–351; MR1311354 (96i:32039)].