ZETA FUNCTIONS AND ` KONTSEVICHINVARIANTS ' ON SINGULAR VARIETIESWillem
نویسنده
چکیده
Let X be a nonsingular algebraic variety in characteristic zero. To an eeective 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 eeective Q{Cartier divisors on X whose support contains the singular locus of X.
منابع مشابه
Zeta functions of nilpotent groups - singular Pfaffians
The local normal zeta functions of a finitely generated, torsion-free nilpotent group G of class 2 depend on the geometry of the Pfaffian hypersurface associated to the bilinear form induced by taking commutators in G. The smallest examples of zeta functions which are not finitely uniform arise from groups whose associated Pfaffian hypersurfaces are plane curves. In this paper we study groups w...
متن کاملMotivic Zeta Functions for Curve Singularities
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring OP,X at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if OP,X is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré se...
متن کاملSpectral analysis and zeta determinant on the deformed spheres
We consider a class of singular Riemannian manifolds, the deformed spheres S k , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian ∆SN k , we study the associated zeta functions ζ(s,∆SN k ). We introduce ...
متن کامل2 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 (Igus...
متن کاملRenormalization of Multiple Zeta Values
Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually singular. We defin...
متن کامل