Foliage — Feuilletages et géométrie algébrique

Projet ANR-16-CE40-0008 de l'Agence nationale de la recherche (2016-2020)

Foliations over algebraic varieties defined over number fields: transcendence proofs and zero lemmas

Ile de Tatihou, 2–6 juillet 2018


1) Arithmetic geometry for complex geometers: from fibrations to arithmetic schemes (3 hours; Carlo Gasbarri)

These lectures will present some basic notions of Diophantine geometry — such as models over the integers of quasi-projective varieties over a number field, hermitian vector bundles, Arakelov degree, or heights — that play a central role in transcendence proofs, in a manner that will emphasize the analogy with the geometric notions of varieties fibered over over curve, of vector bundles, and of degree of a projective variety with respect to some line bundle, familiar to complex geometers.

2) Transcendence for algebraic and analytic geometers (9 hours; Jean-Benoît Bost)

I will give present some classical transcendence theorems in a way that will emphasize their interpretation as algebraization theorems concerning some combination of formal data over the integers and complex analytic data, and their analogy with some "geometric" algebraization theorems, such as Chow’s theorem or Lefschetz’ theorems à la Grauert-Grothendieck, concerning analytic or formal data.

In the first part of the lectures, I will focus on the theorem of Schneider-Lang, and present a proof of this theorem formally parallel to some simple proof of Chow’s theorem. In a second part, I will discuss some transcendence results — such as Baker’s results on "linear forms in logarithms" and the theorem of Siegel-Shidlovski — the proofs of which crucially involve some zero lemma.

Foliations on algebraic varieties over number fields and the algebraicity properties of their leaves will be a unifying theme behind the various transcendence results I will discuss.

3) Zero estimates for solutions of differential equations (6 hours ; Gal Binyamini) CANCELLED

I will discuss the problem of estimating the order of zero of a polynomial restricted to the trajectory of a polynomial vector field. In particular I will present the topological approach of Gabrielov and the algebraic approach of Nesterenko, and some more recent variations. Depending on the interests of the audience I will also discuss some related questions, including:
- The relation between zero estimates and a problem in differential algebra; and applications for the geometry of transcendental numbers (following Pillay and Hrushovski).
- The generalization of zero estimates to foliations of dimension greater than one (following Gabrielov-Khovanskii and other more recent results).
- Applications of zero estimates around the Pila-Wilkie theorem and Wilkie conjecture on counting rational points in transcendental sets.

4) Zeroes and rational points of smooth functions (2 hours ; Georges Comte) SLIDES

I will present several new results concerning the counting of zeroes and rational points on smooth curves, encompassing the case of curves parameterized by elementary functions (possibly oscillating), and the case of graphs of holomorphic and meromorphic functions.

5) Recent developments and perspectives (2 hours ; Tiago Jardim da Fonseca)


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Jean-Benoît Bost, Carlo Gasbarri, Frank Loray, Erwan Rousseau