Germs of arcs on singular algebraic varieties and motivic integration

  title={Germs of arcs on singular algebraic varieties and motivic integration},
  author={Jan Denef and François Loeser},
  journal={Inventiones mathematicae},
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… 


  • Q. Lê
  • Mathematics
    Dalat University Journal of Science
  • 2022
This paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These equivalences lead to

Voevodsky Motives, Stable Homotopy Theory, and Integration

In this thesis, we study applications and connections of Voevodsky’s theory of motives to stable homotopy theory, birational geometry, and arithmetic. On the one hand, we show that we can use the

Equivariant motivic integration and proof of the integral identity conjecture for regular functions

We develop Denef–Loeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the

Integration of Voevodsky motives

In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge

Motivic Milnor fibers of plane curve singularities

We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has

La fibration de Hitchin-Frenkel-Ngo et son complexe d'intersection

In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup.

Motivic invariant of real polynomial functions and Newton polyhedron

We propose a computation of real motivic zeta functions for real polynomial functions, using Newton polyhedron. As a consequence we show that the weights are blow-Nash invariants of convenient

Part VII. An introduction to $p$-adic and motivic zeta functions and the monodromy conjecture

These notes give a basic introduction to the theory of $p$-adic and motivic zeta functions, motivic integration, and the monodromy conjecture.

Symmetries of some motivic integrals

We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational

Generically finite morphisms and formal neighborhoods of arcs

Let f : X → Y be a morphism of pure-dimensional schemes of the same dimension, with X smooth. We prove that if $${\gamma\in J_{\infty}(X)}$$ is an arc on X having finite order e along the



Uniform p-adic cell decomposition and local zeta functions.

The purpose of this paper is to give a cell decomposition for p-adic fields, uniform in p. This generalizes a cell decomposition for fixed p, proved by Denef [7], [9]. We also give some applications

Rational points in henselian discrete valuation rings

© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://

Diophantine Problems Over Local Fields: III. Decidable Fields

In [3] we gave a complete set of elementary axioms for the valued field of p-adic numbers. In this paper we show how the valued field F((t)) of formal power series over a field F (of characteristic

Arc structure of singularities

Introduction. This paper is motivated as much by some interesting possible truths which were encountered as by the things which could be established. Perhaps others will be able to complete the

Un critère d'extension d'un foncteur défini sur les schémas lisses

Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a

Courbes trac'ees sur une germe d''hypersurface

Soit k un corps, A une k-algebre locale, noetherienne, complete pour la topologie definie par son ideal maximal M et (V, v) = Spec A le germe de k-variete algebroide associee. Dans tout ce qui suit,

Éléments de géométrie algébrique

© Publications mathématiques de l’I.H.É.S., 1965, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://