Motivic Zeta Functions Research Articles

The motivic real Milnor fibres

Given a polynomial with real coefficients, we produce a motivic analog of a simple identity that relates the complex conjugation and the monodromy of the Milnor fibre of its complexification. To that purpose, we introduce motivic zeta functions that take into account complex conjugation and monodromy.

Motivic zeta functions of abelian varieties, and the monodromy conjecture

We prove for abelian varieties a global form of Denef and Loeserʼs motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety A over a complete discretely valued field with algebraically closed residue field, its motivic zeta function has a unique pole at Chaiʼs 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 ℓ in C , the value exp ( 2 π i c ( A ) ) is an ℓ-adic tame monodromy eigenvalue of A. The main tool in the paper is Edixhovenʼs filtration on the special fiber of the Néron model of A, which measures the behavior of the Néron model under tame base change.

ALGEBRO-GEOMETRIC FEYNMAN RULES

We give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining "motivic Feynman rules". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.

Motivic zeta functions for curve singularities

AbstractLetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

Motivic zeta functions for curve singularities

AbstractLetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

Zeta functions in triangulated categories

We prove the 2-out-of-3 property for the rationality of the motivic zeta function in distinguished triangles in Voevodsky’s category Open image in new window . As an application, we show the 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 Open image in new window . Together with a result due to P. O’sullivan, this also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.

Zeta Functions and Motives

We study properties of rigid K-linear ⊗-categories A, where K is a field of characteristic 0. When A is semi-simple, we introduce a notion of multiplicities for an object of A: they are rational integers in important cases including that of pure numerical motives over a field. This yields an alternative proof of the rationality and functional equation of the zeta function of an endomorphism, and a simple proof that the number of rational points modulo q of a smooth projective variety over Fq only depends on its “birational motive”. The multiplicities of motives of abelian type over a finite field are equal to ±1. We also study motivic zeta functions, and an abstracted version of the Tate conjecture over finite fields.

Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions

AbstractIn this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the e-invariant of such a ruled exceptional surface E is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of E). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of e to the study of the poles of the well-known topological, Hodge and motivic zeta functions.

Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫ | f | 2 s ω , where the ω are C ∞ differential forms with compact support, and eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form exp ( 2 π −1 s 0 ) , where s 0 is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.

Motivic invariants of arc-symmetric sets and blow-Nash equivalence

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

On the Geometric Determination of the Poles of Hodge and Motivic Zeta Functions

On the Geometric Determination of the Poles of Hodge and Motivic Zeta Functions

Nearby cycles and composition with a nondegenerate polynomial

Let Xj be smooth varieties over a field k of characteristic zero, for 1 ≤ j ≤ p. Consider a family f of p functions fj : Xj → Ak. We will denote also by fj the function on the product X = ∏ j Xj obtained by composition with the projection. We denote by X0(f) the set of common zeroes in X of the functions fj. Let P ∈ k[y1, . . . , yp] be a polynomial, which we assume to be nondegenerate with respect to its Newton polyhedron. In the present paper, we will compute the motivic nearby cycles SP(f) on X0(f) of the composed function P(f) on X as a sum over the set of compact faces δ of the Newton polyhedron of P. For every such δ, we denote by Pδ the corresponding quasihomogeneous polynomial. We associate to such a quasihomogeneous polynomial a convolution operator ΨPδ , which in the special case where Pδ is the polynomial Σ = y1 + y2 is nothing but the operator ΨΣ considered in [9]. For such a compact face δ, one may also define generalized nearby cycles S f , constructed as the limit, as T → ∞, of certain truncated motivic zeta functions. Our main result, Theorem 3.2, follows from additivity from the following statement, Theorem 3.3:

Rationality criteria for motivic zeta functions

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 the Kodaira dimension of the surface is negative.

Motivic zeta functions of infinite-dimensional Lie algebras

The paper shows how to associate a motivic zeta function with a large class of infinite dimensional Lie algebras. These include loop algebras, affine Kac-Moody algebras, the Virasoro algebra and Lie algebras of Cartan type. The concept of a motivic zeta functions provides a good language to talk about the uniformity in p of local p-adic zeta functions of finite dimensional Lie algebras. The theory of motivic integration is employed to prove the rationality of motivic zeta functions associated to certain classes of infinite dimensional Lie algebras.

On motivic zeta functions and the motivic nearby fiber

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.

Geometric determination of the Poles of Highest and second Highest order of Hodge And motivic Zeta functions

AbstractTo any f ∈ ℂ[x1, … ,xn] \ ℂ with f(0) = 0 one can associate the motivic zeta function. Another interesting singularity invariant of f-1{0} is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for n = 2 a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order n − 1 and n. We also show how to obtain the same results for the motivic zeta function.

Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces

To any polynomial f∈ℂ[x 1,…,x n ]∖ℂ with f(0)=0 one associates the well-known motivic zeta function and its specialization to the level of Hodge polynomials. These zeta functions can be given very explicitly in terms of an embedded resolution of f −1{0} in 𝔸ℂ n . In this paper, where we work with polynomials in three variables, i.e., n=3, we find a geometric condition for having a nonzero contribution to the residue at a candidate pole. More precisely, for a given embedded resolution h we fix an exceptional surface E with h(E)={0}, which induces in a canonical way a candidate pole q of the motivic zeta function. Then we prove that, when the surface E is non-rational and we are in a generic situation, the maximality of the logarithmic Kodaira dimension of E ŝ implies the non-vanishing of the contribution of E to the residue at q. Here E ŝ denotes the part of E that doesn't belong to any other irreducible component of h −1(f −1{0}). The same result is already true on the level of Hodge polynomials.

Motivic-type invariants of blow-analytic equivalence

To a given analytic function germ f:(ℝ d ,0)→(ℝ,0), we associate zeta functions Z f,+ , Z f,- ∈ℤ[[T]], defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of Brieskorn polynomials of three variables.

Motivic zeta functions for prehomogeneous vector spaces and castling transformations

AbstractWe study the behaviour of motivic zeta functions of prehomogeneous vector spaces under castling transformations. In particular we deduce how the motivic Milnor fibre and the Hodge spectrum at the origin behave under such transformations.