Motivic Zeta Functions Research Articles

Higher Jacobian ideals, contact equivalence and motivic zeta functions

We show basic properties of higher Jacobian matrices and higher Jacobian ideals for functions and apply it to obtain two main results concerning singularities of functions. Firstly, we prove that a higher Nash blowup algebra is invariant under contact equivalences, which was recently conjectured by Hussain, Ma, Yau and Zuo. Secondly, we obtain an analogue of a result on motivic nearby cycles by Bussi, Joyce and Meinhardt.

Motivic zeta functions of the Hilbert schemes of points on a surface

Let K be a discretely-valued field. Let X\rightarrow\mathrm{Spec}K be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes \mathrm{Hilb}^{n}(X) over the ring of integers R\subseteq K . We exploit this construction in order to compute the motivic zeta function of \mathrm{Hilb}^{n}(X) in terms of the motivic zeta functions of X and of its base-changes with respect to the totally ramified extensions of K . We determine the poles of Z_{\mathrm{Hilb}^n(X)} and study its monodromy property, showing that if the monodromy conjecture holds for X then it holds for \mathrm{Hilb}^{n}(X) too.

Motivic zeta functions of hyperplane arrangements

AbstractFor each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

Monodromy conjecture for semi‐quasihomogeneous hypersurfaces

AbstractWe give a proof of the monodromy conjecture relating the poles of motivic zeta functions with roots of b‐functions for isolated quasihomogeneous hypersurfaces, and more generally for semi‐quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms.

On the motive of the nested Quot scheme of points on a curve

Let C be a smooth curve over an algebraically closed field k, and let E be a locally free sheaf of rank r. We compute, for every d>0, the generating function of the motives [QuotC(E,n)]∈K0(Vark), varying n=(0≤n1≤⋯≤nd), where QuotC(E,n) is the nested Quot scheme of points, parametrising 0-dimensional subsequent quotients E↠Td↠⋯↠T1 of fixed length ni=χ(Ti). The resulting series, obtained by exploiting the Białynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of C. In particular, it is a rational function.

On the cohomology of certain subspaces of text {Sym}^n(mathbb {P}^1) and Occam\u2019s razor for Hodge structures

Vakil and Matchett-Wood (Discriminants in the Grothendieck ring of varieties, 2013. arXiv:1208.3166) made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of text {Sym}^n(mathbb {P}^1). In this note, we disprove one of them and prove a stronger form of the other, thereby obtaining (counter)examples to the principle of Occam’s razor for Hodge structures.

The motivic zeta functions of a matroid

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincar\'{e} duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer.

Motivic zeta functions on [formula omitted]-Gorenstein varieties

We study motivic zeta functions for Q-divisors in a 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 action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a Q-resolution.

Irrationality of motivic zeta functions

Let K0(VarQ)[1/L] denote the Grothendieck ring of Q-varieties with the Lefschetz class inverted. We show that there exists a K3 surface X over Q such that the motivic zeta function ζX(t):=∑n[SymnX]tn regarded as an element in K0(VarQ)[1/L][[t]] is not a rational function in t, thus disproving a conjecture of Denef and Loeser.

Divisorial Motivic Zeta Functions for Marked Stable Curves

We define a divisorial motivic zeta function for stable curves with marked points, which agrees with Kapranov’s motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational and give a formula in terms of the dual graph of the curve.

Computing motivic zeta functions on log smooth models

We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

Motivic zeta functions of degenerating Calabi–Yau varieties

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.

Euler reflexion formulas for motivic multiple zeta functions

We introduce a new notion of ⧆ \boxast -product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability of 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 functions is nicely formulated to deal with the ⧆ \boxast -product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well-known motivic Thom-Sebastiani theorem.

EQUIVARIANT ZETA FUNCTIONS FOR INVARIANT NASH GERMS

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

Splicing motivic zeta functions

The first part of the paper discusses how to lift the splicing formula of Némethi and Veys to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of Némethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of Némethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We illustrate this with some examples.

A survey of Igusa’s local zeta function

The origin and development of the Igusa local zeta function, by Igusa and others, is presented. In particular, we discuss the various conjectures Igusa made and the notable results that have so far been obtained. We also explain how topological and motivic zeta functions arose from the Igusa local zeta function and present the current status of the analogous conjectures. Igusa's conjecture on exponential sums that are related to his zeta function is described, and along with the progress made towards its resolution.

Zeta functions of curves with no rational points

We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.

Poles of maximal order of motivic zeta functions

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 0. 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 nonarchimedean analytification of the degeneration towards the Kontsevich–Soibelman skeleton.

Chow motives versus noncommutative motives

In this article we formalize and enhance Kontsevich’s beautiful insight that Chow motives can be embedded into noncommutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the notions of Schur and Kimura finiteness admit an adequate extension to the realm of noncommutative motives; (2) Gillet–Soulé’s motivic measure admits an extension to the Grothendieck ring of noncommutative motives; (3) certain motivic zeta functions admit an intrinsic construction inside the category of noncommutative motives.

Rationality of motivic zeta functions for curves with finite abelian group actions

Let VarkG 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 VarkG 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 ∑n=0∞[Symn(X,σ)]tn with coefficients in the Grothendieck ring. The main result of this paper asserts that the motivic zeta-function for an algebraic curve with a finite abelian group action is rational. This is a partial generalization of Weil’s First Conjecture.