Julien Sebag

Motivic integration on smooth rigid varieties and invariants of degenerations

We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre’s invariant for p-adic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of Calabi-Yau varieties, our results take a stronger form.

Motivic Serre invariants, ramification, and the analytic Milnor fiber

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

Greenberg Approximation and the Geometry of Arc Spaces

We study the differential properties of generalized arc schemes and geometric versions of Kolchins Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.

Motivic integration and its interactions with model theory and non-Archimedean geometry

Preface 1. Heights and measures on analytic spaces: a survey of recent results, and some remarks Antoine Chambert-Loir 2. C-minimal structures without density assumption Francoise Delon 3. Trees of definable sets in Zp Immanuel Halupczok 4. Triangulated motives over Noetherian separated schemes Florian Ivorra 5. A survey of algebraic exponential sums and some applications Emmanuel Kowalski 6. A motivic version of p-adic integration Karl Rokaeus 7. Absolute desingularization in characteristic zero Michael Temkin.

Introductory notes on the model theory of valued fields

These notes will give some very basic definitions and results from model theory. They contain many examples, and in particular discuss extensively the various languages used to study valued fields. They are intended as giving the necessary background to read the papers by Cluckers, Delon, Halupczok, Kowalski, Loeser and Macintyre in this volume. We also mention a few recent results or directions of research in the model theory of valued fields, but omit completely those themes which will be discussed elsewhere in this volume. So for instance, we do not even mention motivic integration. People interested in learning more model theory should consult standard model theory books. For instance: D. Marker, Model Theory: an Introduction, Graduate Texts in Mathematics 217, Springer-Verlag New York, 2002; C.C. Chang, H.J. Keisler, Model Theory, North-Holland Publishing Company, Amsterdam 1973; W. Hodges, A shorter model theory, Cambridge University Press, 1997.

Differential Equations and A-Approximations

In this article, we introduce the notion of A-approximations associated with a polynomial differential equation F = 0 of order n ≥ 1 and degree d ≥ 2, defined over a differential field of characteristic zero. We also give applications of this construction to the irreducible decomposition of perfect differential ideals, generated by a single element.

Structure Theorems for Greenberg Schemes

Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal \(\mathfrak{m}\) and residue field k. For every integer n⩾0, we set \(R_{n} = R/\mathfrak{m}^{n+1}\).

Prologue: p-Adic Integration

Motivic integration and some of its applications take they very inspiration from results of p-adic integration, that is, integration on analytic manifolds over non-Archimedean locally compact fields.

THE DRINFELD–GRINBERG–KAZHDAN THEOREM IS FALSE FOR SINGULAR ARCS

In this note, we prove that the Drinfeld–Grinberg–Kazhdan theorem on the structure of formal neighborhoods of arc schemes at a nonsingular arc does not extend to the case of singular arcs.

The minimal formal models of curve singularities

Let k be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over k). This is a noetherian affine adic formal k-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive k-parametrization. For the plane curve A2n-singularity, we show that this invariant is Spf(k[[Z]]/〈Zn+1〉). We also obtain information on the minimal formal model of the so-called generalized cusp. We introduce various questions in the direction of the study of these minimal formal models with respect to singularity theory. Our results provide the first positive elements of answer. As a direct application of the former results, we prove that, in general, the isomorphisms satisfying the Drinfeld–Grinberg–Kazhdan theorem on the structure of the formal neighborhoods of arc schemes at non-degenerate arcs do not come from the jet levels. In some sense, this shows that the Drinfeld–Grinberg–Kazhdan theorem is not a formal consequenc...