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Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case. (English) Zbl 1193.14006

The monodromy conjecture for a local analytic isolated singularity \(f: ({\mathbb C}^n, 0) \rightarrow ({\mathbb C}, 0)\) predicts that a pole of its topological zeta function induces one of its monodromy eigenvalues. However, in general, only a few eigenvalues are obtained in this way. The second author of this paper proposed to consider local topological zeta functions \(Z(f, \omega;s)\) associated with \(f\) and with a set of analytic differential \(n\)-forms \(\omega\) living in \(({\mathbb C}^{n}, 0)\). For arbitrary \(n\), he proved [Adv. Math. 213, No. 1, 341–357 (2007; Zbl 1129.14005)], that any given monodromy eigenvalue of \(f\) at \(0\) is induced by a pole of a local topological zeta function of \(f\) and a suitable \(\omega\). In particular for \(n=2\), if \( \lambda\) is a monodromy eigenvalue of \(f\) at \(0\), then there exists a differential 2-form \(\omega\) such that \(Z(f,\omega;s)\) has a pole \(s_0\) satisfying exp\((2 \pi i s_0)=\lambda\). The zeta functions of this result will, in general, have other poles that do not induce monodromy eigenvalues of \(f\).
In this interesting paper, the authors consider an irreducible complex plane curve singularity \(\{f=0\}\) with \(r\) Puiseux pairs and the goal of the paper is to find an infinite set \({\mathcal I}\) of indices \(I \in {\mathbb Z}^{r+1}\) , providing differential forms \(\omega_I\), such that it holds:
(1) If \(s_0\) is a pole of some \(Z(f,\omega_I;s)\), then exp\((2 \pi i s_0)\) is an eigenvalue of monodromy of \(f\).
(2) If \(\lambda\) is an eigenvalue of monodromy of \(f\), then there exists a form \(\omega_I\) and a pole \(s_0\) of \(Z(f,\omega_I;s)\) such that exp\((2 \pi i s_0)=\lambda\).
Furthermore, in the paper it is proved that there exists a finite subset \({\mathcal J}\) of \({\mathcal I}\) such that both sets \({\mathcal J}\) and \({\mathcal I}\) satisfy conditions (1) and (2).

MSC:

14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings
32S05 Local complex singularities
14H50 Plane and space curves

Citations:

Zbl 1129.14005
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