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.
We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…
This is a survey on motivic zeta functions associated to abelian varieties and Calabi-Yau varieties over a discretely valued field. We explain how they are related to Denef and Loeser's motivic zeta function associated to a complex hypersurface singularity and we investigate the relation between the poles of the zeta function and the eigenvalues of the monodromy action on the tame $\ell$-adic cohomology of the variety. The motivic zeta function allows to generalize many interesting arithmetic invariants from abelian varieties to Calabi-Yau varieties and to compute them explicitly on a model wi…
We consider a conjecture of Kontsevich and Soibelman which is regarded as a foundation of their theory of motivic Donaldson-Thomas invariants for non-commutative 3d Calabi-Yau varieties. We will show that, in some certain cases, the answer to this conjecture is positive.
We study aspects related to Kontsevich's homological mirror symmetry conjecture in the case of Calabi-Yau complete intersections in toric varieties. In a 1996 lecture at Rutgers University, Kontsevich indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our main results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories.
In this paper, it is demonstrated that derived equivalence between smooth, projective, connected schemes that are either surfaces or abelian implies equality of zeta functions.
We show that any derived equivalent smooth, projective varieties of dimension 3 over a finite field $\mathbb{F}_q$ have equal zeta functions. This result is an application of the extension to smooth, projective varieties over any field of Popa and Schnell's proof that derived equivalent smooth, projective varieties over $\mathbb{C}$ have isogenous Albanese torsors; this result is proven in an appendix by Achter, Casalaina-Martin, Honigs and Vial.
We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.
Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.
The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of certain motivic cohomology groups. Progress on his conjecture allows one to give a beautiful restatement of the main theorem of the article in terms of $Z$-cohomology groups.
The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities, introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties, and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the Minimal Model Program, we introduce very general stringy invariants, associated to 'almost all' singularities, more precisely to all singularities which are not strictly log canonical. They specialize to the …
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the th…
We prove a 1999 conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of Denef and Loeser's motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi-Yau varieties. Our results suggest that the weight function induces a flow on the non-archimedean analytification of the degeneration towards the Kontsevich-Soibelman skeleton.
We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and c…
We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety $A$ over a complete discretely valued field, its motivic zeta function has a unique pole at Chai's base change conductor $c(A)$ of $A$, and that the order of this pole equals one plus the potential toric rank of $A$. Moreover, we show that for every embedding of $\Q_\ell$ in $\C$, the value $\exp(2πi c(A))$ is an $\ell$-adic tame monodromy eigenvalue of $A$. The main tool in the paper is Edixhoven's…
We provide a formula for the generating series of the zeta function $Z(X,t)$ of symmetric powers $Sym^n X$ of varieties over finite fields. This realizes $Z(X,t)$ as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in $W(R)$ the big Witt ring of $R=W(\mathbb{Z})$. We apply our formula to compute $Z(Sym^n X,t)$ in a number of explicit cases. Moreover, we show that all $λ$-ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for $Z(X,t)$ takes the form of a MacDonald formula for the zeta function.
Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,σ)$, where $X$ is a variety over $k$ and $σ$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,σ)]t^n$ with coefficients in the Grothendieck rin…
We prove that if $X$ is a smooth projective variety of dimension greater than 1 over a field $K$ of characteristic zero such that $\operatorname{Pic}(X_{\bar{K}}) = \mathbb{Z}$ and $X_{\bar{K}}$ is simply connected, then the natural map $ρ: \operatorname{Aut}(X) \to \operatorname{Aut}(\operatorname{Sym}^d(X))$ is an isomorphism for every $d > 0$. We also partially compute the motivic zeta function of a Severi-Brauer surface and explain some relations between the classes of Severi-Brauer varieties in the Grothendieck ring of varieties.
We present a method for computing the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. We have implemented our approach for some surfaces using the Magma programming language, and present some explicit examples which we have computed.
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the property of having a rational zeta function satisfying a functional equation is preserved under products.
The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is rational if and only if its Kodaira dimension is negative.
We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, …
Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case…
We collect some properties of the motivic zeta functions and the motivic nearby fiber defined by Denef and Loeser. In particular, we calculate the relative dual of the motivic nearby fiber. We give a candidate for a nearby cycle morphism on the level of Grothendieck groups of varieties using the motivic nearby fiber.
We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.
The motivic Hilbert zeta function of a variety is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points of the variety. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.
We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second group $\bar{Spin(A)}$ is defined as follows: the group of autoequivalences of the bounded derived category of coherent sheaves acts on the total cohomology $H(A,Q)$ of an abelian variety $A$ via algebraic correspondences. The group $\bar{Spin(A)}$ is now the Zariski closure of its image in $GL(H(A,Q))$. Our constructions are compatible with the picture of mirror…
We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the …
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function, which instead respects the tensor product of motives. There is no analogue of the Weil conjectures, and we give a sufficient criterion for an analytic continuation to exist. This happens, for example, for cellular varieties, abelian varieties, or genus g > 1 curves with a supersingular Jacobian.
We introduce a formal integral on the system of varieties mapping properly and birationally to a given one, with value in an associated Chow group. Applications include comparisons of Chern numbers of birational varieties, new birational invariants, `stringy' Chern classes, and a `celestial' zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a `change of variable' formula holds. The formalism has points of contac…
Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by nai…
We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the size of the maximal Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.
We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $\mathbb{S} \to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $π_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theor…
In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:140…
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula …
Let $K_0(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}]$ denote the Grothendieck ring of $\mathbb{Q}$-varieties with the Lefschetz class inverted. We show that there exists a K3 surface X over $\mathbb{Q}$ such that the motivic zeta function $ζ_X(t) := \sum_n [\mathrm{Sym}^n X]t^n$ regarded as an element in $K_0(\mathrm{Var}_{\mathbb{Q}})[1/\mathbb{L}][[t]]$ is not a rational function in $t$, thus disproving a conjecture of Denef and Loeser.
The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^{m}. In particular we extend Igusa's method for estimating exponential su…
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko and Varchenko etc. concerning polynomials on $\CC^n$ will be generalized to polynomial functions on any toric variety.
We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the $\boxast$-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta…
We prove: if the (étale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives comi…
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of mot…
We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.
Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\mathrm{char}\, k$ we give a classification of group schemes $B[\ell]$, where $B$ runs through the isogeny class, in terms of certain Newton polygons associated to $f_A$. As an application we classify zeta functions of Kummer surfaces over $k$.
We introduce the etale framework to study Igusa zeta functions in several variables, generalizing the machinery of vanishing cycles in the univariate case. We define the etale Alexander modules, associated to a morphism of varieties F from X to affine r-space, a geometric point of X, and an object in the derived category of constructible l-adic sheaves on the inverse image of the r-torus under F. The Alexander modules are sheaves of modules on the scheme of continuous characters of the tame fundamental group of the r-torus. We formulate Loeser's Monodromy and Holomorphy Conjectures for multiva…
We establish a trace formula for rigid varieties $X$ over a complete discretely valued field, which relates the set of unramified points on $X$ to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring $R$, and we introduce the Weil generating series of a regular formal $R$-scheme $\mathfrak{X}$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When $\mathfr…
This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in P^n. In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the p-adic Picard-Fuchs equation associated with monomial deformations of Fermat hypersurfaces. As a by-product we obtain Poincare Duality for the Rigid cohomology of certa…
We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in DM. Joint with a result of P.O'Sullivan it also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
