Zeta functions and 'Kontsevich invariants' on singular varieties
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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 Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the 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 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.Keywords
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This publication has 13 references indexed in Scilit:
- Motivic Integration, Quotient Singularities and the McKay CorrespondenceCompositio Mathematica, 2002
- Torification and factorization of birational mapsJournal of the American Mathematical Society, 2002
- Non-Archimedean integrals and stringy Euler numbers of log-terminal pairsJournal of the European Mathematical Society, 1999
- THE TOPOLOGICAL ZETA FUNCTION ASSOCIATED TO A FUNCTION ON A NORMAL SURFACE GERMTopology, 1999
- Germs of arcs on singular algebraic varieties and motivic integrationInventiones Mathematicae, 1999
- Zeta Functions for Curves and Log Canonical ModelsProceedings of the London Mathematical Society, 1997
- Determination of the poles of the topological zeta function for curvesmanuscripta mathematica, 1995
- Towards Birational Classification of Algebraic VarietiesBulletin of the London Mathematical Society, 1987
- Complex powers and asymptotic expansions. II.Journal für die reine und angewandte Mathematik (Crelles Journal), 1975
- The topology of normal singularities of an algebraic surface and a criterion for simplicityPublications mathématiques de l'IHÉS, 1961