Willem Veys | KU Leuven (original) (raw)

Papers by Willem Veys

Topology, 1999

We associate to a regular function f on a normal surface germ (S; 0) an invariant, called the top... more We associate to a regular function f on a normal surface germ (S; 0) an invariant, called the topological zeta function, which generalizes the same invariant for a plane curve germ; by de nition it is a rational function in one variable. We study its poles and their relation with the local monodromy of f , in particular we prove the`generalized holomorphy conjecture'. We give a formula for this topological zeta function in terms of the log canonical model of (S; f ?1 f0g), and we also introduce a still more general invariant.

Proceedings of the London Mathematical Society, 1997

The topological zeta function and Igusa's local zeta function are respectively a geometrical inva... more The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p{adic eld. When f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so{called log canonical model of f ?1 f0g in A 2 . This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non{symmetric`q{deformation' of the intersection matrix of the minimal resolution of a Hirzebruch{Jung singularity.

Mathematische Annalen, 1998

We study mainly connected con gurations of irreducible curves r i=1 C i on a nonsingular rational... more We study mainly connected con gurations of irreducible curves r i=1 C i on a nonsingular rational projective complex surface X such that the Euler characteristic (X n r 1991 Mathematics Subject Classi cation. 14J26 14E07.

Mathematische Annalen, 1993

Manuscripta Mathematica, 1995

![Research paper thumbnail of Zeta Functions and `Kontsevich Invariants' on Singular Varieties](https://attachments.academia-assets.com/42287053/thumbnails/1.jpg)

Canadian Journal of Mathematics, 2001

Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kon... more 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 (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 padic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.

Advances in Mathematics, 2007

For a complex polynomial or analytic function f , there is a strong correspondence between poles ... more 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 | 2s 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(2π √ −1s 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.

Proceedings of the American Mathematical Society

We study the topological zeta function Z_{top,f}(s) associated to a polynomial f with complex coe... more We study the topological zeta function Z_{top,f}(s) associated to a polynomial f with complex coefficients. This is a rational function in one variable and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote P_n := {s_0 | \exists f in C[x_1,..., x_n] : Z_{top,f}(s) has a pole in s_0}. We show that {-(n-1)/2-1/i | i in Z_{>1}} is a subset of P_n; for n=2 and n=3, the last two authors proved before that these are exactly the poles less then -(n-1)/2. As main result we prove that each rational number in the interval [-(n-1)/2,0) is contained in P_n.

Pacific Journal of Mathematics, 2001

Let F be a number field and f ∈ F (x1 ,... , xn) \ F.T o any completion K of F and any character ... more Let F be a number field and f ∈ F (x1 ,... , xn) \ F.T o any completion K of F and any character κ of the group of units of the valuation ring of K one associates Igusa&amp;amp;amp;amp;amp;amp;#39;s local zeta function ZK(κ, f, s). The holomorphy conjecture states that for all except a finite number of completions K

Let (S; 0) be a normal surface germ and f a nonconstant regular function on (S; 0)with f(0) = 0. ... more Let (S; 0) be a normal surface germ and f a nonconstant regular function on (S; 0)with f(0) = 0. Using any additive invariant on complex algebraic varieties one canassociate a zeta function to these data, where the topological and motivic zeta functionare the roughest and the nest one, respectively. In this paper we are interested in ageometric determination of

Journal of Approximation Theory, 2008

We consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t = ... more We consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t = 0 in the starting point a and end at time t = 1 in the endpoint b and the other n/2 Brownian motions start at time t = 0 at the point −a and end at time t = 1 in the point −b, conditioned that the n Brownian paths do not intersect in the whole time interval (0, 1). The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case ab < 1/2. We find that the limiting mean density of the positions of the Brownian motions is supported on one or two intervals and that the correlation kernel has the usual scaling limits from random matrix theory, namely the sine kernel in the bulk and the Airy kernel near the edges.

Journal of Algebra, 2007

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a... more Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f = 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension dn/2 . Finally, we look at the consequences for the poles of the motivic zeta function.

International Mathematics Research Notices, 2010

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singul... more Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of 'mild' isolated singularities (the allowed singularities depend on the dimension). As a corollary we obtain a proof of Batyrev's conjecture for threefolds in full generality. In these cases, we also give an explicit description of the stringy Hodge numbers and we suggest a possible generalized definition of stringy Hodge numbers if the E-function is not a polynomial.

International Mathematics Research Notices, 2008

In this article we consider surfaces that are general with respect to a 3dimensional toric ideali... more In this article we consider surfaces that are general with respect to a 3dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces.

Comptes Rendus Mathematique, 2007

... Helv. 50 (1975) 233–248. [2] E. Artal-Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle-Hernánde... more ... Helv. 50 (1975) 233–248. [2] E. Artal-Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle-Hernández, Monodromy conjecture for some surface singularities, Ann. Sc. ... Soc. 5 (4) (1992) 705–720. [6] F. Loeser, Fonctions d'Igusa p-adiques et polynômes de Bernstein, Amer. J. Math. ...

Bulletin of the London Mathematical Society, 1999

The global and local topological zeta functions are singularity invariants associated to a polyno... more The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By de nition these zeta functions are rational functions in one variable and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non degenerate with respect to its Newton polyhedron we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; concerning the global zeta function we give a similar result. Moreover for any f we show that poles of maximal possible order are always of the form ?1=N with N a positive integer.

Bulletin of the London Mathematical Society, 2010

The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topologic... more The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one find a list of differential forms ω i such that any pole of the zeta function of f and an ω i induces a monodromy eigenvalue of f , and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f .

Advances in Geometry, 2000

We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodg... more We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Noguès, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to compute the contribution of a Brieskorn singularity to the stringy E-function. As a corollary, we prove that stringy Hodge numbers of varieties with a certain class of strictly canonical Brieskorn singularities are nonnegative. We conclude by computing an interesting 6-dimensional example. It shows that a result, implying nonnegativity of stringy Hodge numbers in lower dimensional cases, obtained in our previous paper, is not true in higher dimension.

Geometry & Topology, 2012

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimensi... more The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f : (X, 0) → (C, 0) defined on a normal surface singularity (X, 0). The article targets the 'right' extension in the case when the link of (X, 0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f, ω; s) for any f and analytic differential form ω, which will play the key technical localization tool in the later definitions and proofs.