Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces (Q1416706)

The motivic zeta function \(Z_{\text{mot}}(f;s)\) of \textit{J. Denef} and \textit{F. Loeser} [J. Alg. Geom. 7, 505--537 (1998; Zbl 0943.14010)] is essentially a singularity invariant, associated to a non-constant polynomial \(f \in \mathbb C[x_1,\dots,x_n]\). There is an explicit formula in terms of an embedded resolution \(h\) of \(\{f=0\}\) in affine \(n\)-space; we have in particular that each irreducible component of \(h^{-1}\{f=0\}\) induces a candidate pole of \(Z_{\text{mot}}(f;s)\) and that each pole is obtained in this way. For \(n=2\) there is a geometric criterion to decide whether such a candidate pole is really a pole [\textit{W. Veys}, Manuscr. Math. 87, 435--448 (1995; Zbl 0851.14012)]. Finding nice geometric conditions in higher dimensions, assuring that a candidate pole is really a pole, is a difficult problem. The author developed a conceptual condition, valid in arbitrary dimension [J. Reine Angew. Math. (to appear)]. In this paper he proves a geometric criterion of a different flavour for \(n=3\). It is stated roughly as follows. Fix a non-rational exceptional surface \(E\) of the resolution \(h\) which is mapped to a point by \(h\). If the open part of \(E\), that doesn't belong to any other component of \(h^{-1}\{f=0\}\), is of log general type, then generically the candidate pole associated to \(E\) is a pole. The author explains why it is natural to consider this condition of maximal logarithmic Kodaira dimension, in particular by reformulating the result for \(n=2\). More precisely the result is in fact proven for the more manageable and concrete \textsl{Hodge zeta function} of \(f\), which is a priori a stronger result than for the motivic one.