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On the monodromy conjecture for curves on normal surfaces. (English) Zbl 1063.14045

To a normal complex (algebraic) surface germ \((S,0)\) and a non-constant regular function \(f\) on it one associates the so-called topological zeta function. This invariant contains information about both the singularity \((S,0)\) and the possible singularity at \(0\) of the curve \(\{f=0\}\) on \(S\). One is especially interested in the poles of this function, for example in the relation of these poles with the eigenvalues of the local monodromy of \(f\). When \((S,0)\) is nonsingular an important result in this matter is the \` Monodromy Conjecture\', proved by F. Loeser [Am. J. Math. 110, No. 1, 1–21 (1988; Zbl 0644.12007)]. It states that, if a rational number \(s\) is a pole of the topological zeta function of \(f\), then \(e^{2\pi is}\) is an eigenvalue of the local monodromy of \(f\) at some point of \(\{f=0\}\).
In this paper the author first gives his own, rather elementary and conceptual, proof of the conjecture. This is not only included for its simplicity; the author also needs exactly the same arguments in the second part, where he explains in detail what can, and what cannot be expected in the singular case. In particular he proves, by giving an actual counterexample, that not only the analogous statement, but also the by the reviewer in [Topology 38, No. 2, 439–456 (1999; Zbl 0947.32020)] proposed adapted version, does not hold in the general singular setting. These results are also true for the well-known motivic zeta function.
Reviewer: Wim Veys (Leuven)

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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