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Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties
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- Journal:
- Canadian Journal of Mathematics / Volume 53 / Issue 4 / 01 August 2001
- Published online by Cambridge University Press:
- 20 November 2018, pp. 834-865
- Print publication:
- 01 August 2001
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- Article
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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 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
$\mathbb{Q}$-Gorenstein variety $X$ we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective $\mathbb{Q}$-Cartier divisors on $X$ whose support contains the singular locus of $X$.
