2 00 0 Zeta Functions and ‘ Kontsevich Invariants ’ on Singular Varieties

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 Grotendieck 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 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 Q–Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective Q–Cartier divisors on X whose support contains the singular locus of X.