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Zeta Functions and Kontsevich Invariants' on Singular Varieties

Published:2001-08-01
Printed: Aug 2001
• Willem Veys
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Abstract

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant {\it motivic (Igusa) zeta function}, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\bbQ$-Gorenstein variety $X$ we associate a motivic zeta function and a `Kontsevich invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support contains the singular locus of~$X$.
 Keywords: singularity invariant, topological zeta function, motivic zeta function
 MSC Classifications: 14B05 - Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14E15 - Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 32S50 - Topological aspects: Lefschetz theorems, topological classification, invariants 32S45 - Modifications; resolution of singularities [See also 14E15]

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