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

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Abstract.

To any polynomial f∈ℂ[x 1,…,x n ]∖ℂ with f(0)=0 one associates the well-known motivic zeta function and its specialization to the level of Hodge polynomials. These zeta functions can be given very explicitly in terms of an embedded resolution of f −1{0} in 𝔸ℂ n. In this paper, where we work with polynomials in three variables, i.e., n=3, we find a geometric condition for having a nonzero contribution to the residue at a candidate pole. More precisely, for a given embedded resolution h we fix an exceptional surface E with h(E)={0}, which induces in a canonical way a candidate pole q of the motivic zeta function. Then we prove that, when the surface E is non-rational and we are in a generic situation, the maximality of the logarithmic Kodaira dimension of E ŝ implies the non-vanishing of the contribution of E to the residue at q. Here E ŝ denotes the part of E that doesn't belong to any other irreducible component of h −1(f −1{0}). The same result is already true on the level of Hodge polynomials.

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Authors and Affiliations

  1. Departement Wiskunde, Celestijnenlaan 200B, 3001, Leuven, Belgium

    B. Rodrigues

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  1. B. Rodrigues
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Correspondence to B. Rodrigues.

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Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium).

Mathematics Subject Classificaton (2000): 14B05, 14E15, 14J17 (32S45)

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Rodrigues, B. Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces. manuscripta math. 112, 137–159 (2003). https://doi.org/10.1007/s00229-003-0399-8

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  • DOI: https://doi.org/10.1007/s00229-003-0399-8

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