Ma recherche se situe dans le domaine de la géométrie algébrique.
Plus particulièrement, j'aime appliquer la géométrie analytique non archimédienne, spécialement du point de vue de la théorie de Berkovich, à des problèmes de géométrie birationnelle (théorie des singularités, géométrie Lipschitz, intégration motivique), géométrie arithmétique (modèles de courbes et ramification) ou combinatoire (géométrie tropicale).
Any subanalytic germ (X,0) in (ℝn,0) is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ (X,0).
The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade.
In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces.
We conclude the manuscript with a list of open questions which we believe to be worth of interest.
Publications
"Triangulations of non-archimedean curves, semi-stable reduction, and ramification" (with D. Turchetti)
– Annales de l'Institut Fourier, 73(2), 695–746, 2023 (arXiv, DOI)
Let K be a complete discretely valued field with algebraically closed residue field and let C be a smooth projective and geometrically connected algebraic K-curve of genus g.
Assume that g≥2, so that there exists a minimal finite Galois extension L of K such that CL admits a semi-stable model.
In this paper, we study the extension L|K in terms of the minimal triangulation of Can, a distinguished finite subset of the Berkovich analytification of C.
We prove that the least common multiple d of the multiplicities of the points of the minimal triangulation always divides the degree [L:K].
Moreover, if d is prime to the residue characteristic of K, then we show that d=[L:K], obtaining a new proof of a classical theorem of T. Saito on tame ramification.
We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case.
In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito's theorem to the wildly ramified case.
"Polar exploration of complex surface germs" (with A. Belotto da Silva, A. Némethi, and A. Pichon)
– Transactions of the American Mathematical Society, 375(9), 6747–6767, 2022 (arXiv, DOI)
We prove that the topological type of a normal surface singularity (X,0) provides finite bounds for the multiplicity and polar multiplicity of (X,0), as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of (X,0).
A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of (X,0), which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of (X,0) through its Nash transform.
This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted.
"On Lipschitz normally embedded complex surface germs" (avec A. Belotto da Silva et A. Pichon)
– Compositio Mathematica, 158(3), 623–653, 2022 (arXiv, DOI)
We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs.
We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities.
In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.
"Inner geometry of complex surfaces: a valuative approach" (avec A. Belotto da Silva et A. Pichon)
– Geometry & Topology, 26(1), 163–219, 2022 (arXiv, DOI)
Bonus : voici les transparents d'un exposé de 15 minutes sur le résultat principal de ce papier.
Étant donné un germe analytique complexe (X,0) dans (ℂn,0), la métrique Hermitienne standard de ℂn induit naturellement une métrique par longueurs d’arcs sur (X,0), appelée métrique interne. Nous étudions la structure métrique interne d'un germe de singularité isolée de surface $(X,0)$ via une famille infinie d'invariants numériques appelés taux internes.
Notre résultat principal est une formule calculant le Laplacien de la fonction des taux internes sur un espace de valuations, l'entrelacs non archimédien de (X,0).
Nous en déduisons en particulier que la donnée globale de la topologie de (X,0), avec la configuration d'une section hyperplane générique et de la courbe polaire d'une projection plane générique de (X,0), détermine complètement tous les taux internes de (X,0), et donc la structure métrique locale du germe. Nous décrivons également plusieurs autres applications de notre formule.
In this paper we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map f: 𝔸ℂd → 𝔸ℂ1.
We show that the motivic nearby cycles at infinity Sf,a∞ of f for a value a is a motivic generalization of the classical invariant λf(a), an integer that measures a lack of equisingularity at infinity in the fiber f-1(a).
We then introduce a non-archimedean analytic nearby fiber at infinity Ff,a∞ whose motivic volume recovers the motive Sf,a∞.
With Sf,a∞ and Ff,a∞ can be naturally associated a motivic and an analytic bifurcation sets respectively; we show that the first one always contains the second, and that both contain the classical topological bifurcation set of f if f has isolated singularities at infinity.
"Links of sandwiched surface singularities and self-similarity" (avec C. Favre et M. Ruggiero) – Manuscripta Mathematica, 162(1-2), 23–65, 2020 (arXiv, DOI)
We characterize sandwiched singularities in terms of their link in two different settings.
We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links.
We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs.
We then show that a complex surface singularity is sandwiched if and only if its complex link can be embedded in a Kato surface in such a way that its complement remains connected.
"Galois descent of semi-affinoid spaces" (avec D. Turchetti) – Mathematische Zeitschrift, 290(3-4), 1085–1114, 2018
(arXiv, DOI)
We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a K-analytic space X, provided that X⊗KL is semi-affinoid for some finite tamely ramified extension L of K. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.
"Normalized Berkovich spaces and surface singularities" – Transactions of the American Mathematical Society, 370(11), 7815–7859, 2018
(arXiv, DOI)
We define normalized versions of Berkovich spaces over a trivially valued field k, obtained as quotients by the action of ℝ>0 defined by rescaling semivaluations. We associate such a normalized space to any special formal k-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed G-topological space, which we prove to be G-locally isomorphic to a Berkovich space over the field k((t)) with a t-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of k-varieties, and allow to study the birational geometry of k-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field k is analogous to the structure of non-archimedean analytic curves over k((t)), and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.
"Faithful realizability of tropical curves" (avec M. Cheung, J. Park et M. Ulirsch) – International Mathematics Research Notices, 2016(15), 4706–4727, 2016 (arXiv, DOI)
We study whether a given tropical curve Γ in ℝn can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by Γ. We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph G with rational edge lengths there exists a smooth algebraic curve in a toric variety whose analytification has skeleton G, and the corresponding tropicalization is faithful. Our approach is based on a combination of the theory of toric schemes over discrete valuation rings and logarithmically smooth deformation theory, expanding on a framework introduced by Nishinou and Siebert.
This note announced some of the results of the paper "Normalized Berkovich spaces and surface singularities".
Enseignement
MAA306 - Differential Geometry (2023-2024) sur Moodle.
MAT451 - Algèbre et Théorie de Galois (2023-2024) sur Moodle.
Je suis le Conseiller Académique en charge du volet mathématique de la première année du Bachelor of Science de l'École polytechnique.
Les étudiants qui cherchent des informations peuvent se connecter à Moodle, SynapseS ou me contacter par courriel.
Curriculum Vitae
2021-aujourd'hui: Professeur Monge
École polytechnique.
2019-2021 : Chercheur Humboldt
Goethe Universität Frankfurt (Allemagne), dans le groupe de Annette Werner.
Tropical Geometry in Frankfurt (TGiF), une série d'exposés autour de la géométrie tropicale que je co-organisais avec Martin Ulirsch, est pour le moment resté un séminaire virtuel.
Plus d'informations peuvent se trouver ici.
Voici une ancienne liste de séminaires ayant lieu à Paris ou à proximité :
Et voici des événements intéressants à venir.
Je vais probablement participer seulement à un partie d'entre eux.
Calendrier des événements au CIRM, Luminy : 2023, 2024.
ICM videos: Berlin 1998, Beijing 2002, Madrid 2006, and Hyderabad 2010 are here, Seoul 2014 here (direct YouTube link here, prize lectures here), Rio de Janeiro 2018 here (direct YouTube link here)
Voici quelques vieux articles que j'ai eu du mal à trouver en ligne ou que j'ai numérisés moi-même.
J'espère que les moteurs de recherche les trouveront et que cela fera gagner du temps à d'autres personnes.
polytechnique.edu 