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The monodromy conjecture for plane meromorphic germs. (English) Zbl 1348.14008

In the last decades there is an intense interest in different versions and generalizations of the Monodromy Conjecture of Denef and Loeser. The present manuscript verifies a new case, namely the case of plane meromorphic germs. The notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced by S. M. Gusein-Zade et al. [Funct. Anal. Appl. 32, No. 2, 93–99 (1998; Zbl 0933.32003); translation from Funkt. Anal. Prilozh. 32, No. 2, 26–35 (1998)]. In the present article, the authors extend the definition of the topological zeta function to this case. The main results focus on the computation of the poles in the plane case. The authors prove that the generalization of the monodromy conjecture holds. That is, if \(s_0\) is a pole of the topological zeta function, then \(e^{2\pi i s_0}\) is a monodromy eigenvalue at some point. The article contains several nice examples and comments.

MSC:

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S55 Milnor fibration; relations with knot theory

Citations:

Zbl 0933.32003
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References:

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