The monodromy conjecture for zeta functions associated to ideals in dimension two
Let \(\mathcal{I}=(f_1,\dots,f_r)\) be a nontrivial ideal in \(\mathbb{C}[x_1,\dots,x_n]\) and \(Y=V(\mathcal{I})\) the corresponding subscheme of \(\mathbb{A}_{\mathbb{C}}^{n}\). Assume that \(Y\) contains the origin. Take a log-principalization \(\Psi:\widetilde{X}\rightarrow\mathbb{A}^{n}\) of \(\mathcal{I}\). Let \(\widetilde{E}=\sum_{i\in J}N_{i}E_{i}\) denote the divisor \(\Psi^{\ast }\mathcal{I}\), i.e. its irreducible components are the \(E_{i}\), \(i\in J\), occurring with multiplicity \(N_{i}\). Let the relative canonical divisor of \(\Psi\) be \(\sum_{i\in J}(v_{i}-1)E_{i}\), i.e. \(v_{i} -1\)\ is the multiplicity of \(E_{i}\) in the divisor \(\Psi^{\ast}\left( dx_{1}\wedge\dots dx_{n}\right) \). Set \(E_{I}^{\circ}:=\left( \bigcap_{i\in I}E_{i}\right) \setminus\left( \bigcup_{i\notin I}E_{i}\right) \) for \(I\subset J\). The topological zeta function of \(\mathcal{I}\) at the origin is defined as \[ Z_{top,}\mathcal{I}(s) :=\sum_{I\subset J} \chi\left( E_{I}^{\circ}\cap\Psi^{-1}\{0\}\right) \prod_{i\in I}\frac{1}{v_{i}+N_{i}s}\in\mathbb{Q}(s). \] In the case \(r=1\) the topological zeta function were introduced by \textit{J. Denef} and \textit{F. Loeser} [J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)]. Alternatively these zeta functions can be obtained as a specialization of the motivic zeta functions [cf. \textit{J. Denef} and \textit{F. Loeser}, J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)]. This can be generalized for arbitrary \(r\) [\textit{W. Veys} and \textit{W. A. Zúñiga-Galindo}, Trans. Am. Math. Soc. 360, No. 4, 2205--2227 (2008; Zbl 1222.11141)]. The poles of \(Z_{top, \mathcal{I}}(s)\) belong to the set \(\left\{ \frac{-v_{i} }{N_{i}},i\in J\right\} \). For \(r=1\) there are several intriguing conjectures connecting the poles of \(Z_{top,\mathcal{I}}(s)\) with the eigenvalues of the complex monodromy. In the paper under reviewing the autors, using ``Verdier monodromy'', proved the following version of the generalized monodromy conjecture (see Theorem 4.1): If \(\frac{-v}{N}\) is a pole of the local topological zeta function of an ideal \(\mathcal{I}\subset\mathbb{C}[x,y]\), then there exists a point \(p\in E\) such that \(e^{\frac{-2\pi iv}{N}}\) is an eigenvalue of monodromy in \(p\).
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