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The topological zeta function associated to a function on a normal surface germ. (English) Zbl 0947.32020

To each complex polynomial \(f\) J. Denef and F. Loeser [J. Am. Math. Soc. 5, No. 4, 705-720 (1992; Zbl 0777.32017)] associated a series of topological zeta functions (one for each \(d\in \mathbb N\)), defined in terms of the embedded resolution of \(f^{-1}(0)\). An analogous formula can be written for any function \(f\) on a normal surface germ \((S,0)\). The author extends his previous results for the case \(S\) smooth to this setting. In particular, he gives a formula for the zeta functions in terms of the log-canonical model of the pair \((S,f)\). An example shows that the Monodromy Conjecture does not hold in this case: there exist poles which are not logarithms of eigenvalues of some local monodromy. But the holomorphy conjecture is proved here: if \(d\) does not divide the order of any eigenvalue, then the \(d\)-th zeta function vanishes.
In the final section the author generalises his zeta function to one with coefficients in a certain localisation of the Grothendieck of algebraic varieties, in the spirit of J. Denef and F. Loeser [‘Motivic Igusa zeta functions’, J. Algebraic Geom. 7 (1998)].

MSC:

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14B05 Singularities in algebraic geometry

Citations:

Zbl 0777.32017
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