Lorenzo FANTINI

À propos de moi

Je suis chercheur postdoctoral de la Fondation Alexander von Humboldt au sein de l'équipe Algebra und Geometrie de l'Institut für Mathematik, Goethe-Universität Frankfurt am Main.

News alert! → Je co-organise avec Martin Ulirsch et Annette Werner une conférence sur la géométrie non archimédienne et la géométrie tropicale.
Voici la page web de la rencontre, qui aura lieu à Francfort du 14 au 17 septembre 2020.

Adresse

Recherche

Une courbe de Berkovich à la Mœbius.

Une courbe de Berkovich à la Mœbius.

Publications et prépublications

1. "Triangulations of non-archimedean curves, semi-stable reduction, and ramification" (avec D. Turchetti)
       – soumis, 40 pages, 2019 (arXiv) montrer le résumé
   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 C_L admits a semi-stable model. In this paper, we study the extension L|K in terms of the minimal triangulation of C^an, 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 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.
2. "Inner geometry of complex surfaces: a valuative approach" (avec A. Belotto da Silva et A. Pichon)
       – soumis, 42 pages, 2019 (arXiv) montrer le résumé
   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 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.
3. "Motivic and analytic nearby fibers at infinity and bifurcation sets" (avec M. Raibaut)
       – soumis, 18 pages, 2018 (arXiv) montrer le résumé
   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: 𝔸_ℂ^d → 𝔸_ℂ¹. We show that the motivic nearby cycles at infinity S_f,a^∞ of f for a value a is a motivic generalization of the classical invariant λ_f(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 F_f,a^∞ whose motivic volume recovers the motive S_f,a^∞. With S_f,a^∞ and F_f,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.
4. "Links of sandwiched surface singularities and self-similarity" (avec C. Favre et M. Ruggiero)
       – à paraître dans Manuscripta Mathematica, 35 pages, 2018 (arXiv) montrer le résumé
   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.
5. "Galois descent of semi-affinoid spaces" (avec D. Turchetti)
       – Mathematische Zeitschrift, 290(3), 1085–1114, 2018 (arXiv) montrer le résumé
   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.
6. "Normalized Berkovich spaces and surface singularities"
       – Transactions of the American Mathematical Society, 370(11), 7815–7859, 2018 (arXiv) montrer le résumé
   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.
7. "Faithful realizability of tropical curves" (avec M. Cheung, J. Park et M. Ulirsch)
       – International Mathematics Research Notices, 2016(15), 4706–4727, 2016 (arXiv) montrer le résumé
   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.
8. "Normalized non-archimedean links and surface singularities"
       – Comptes Rendus Mathématique, 352(9), 719–723, 2014 (Pdf) montrer le résumé
   This note announced some of the results of the paper "Normalized Berkovich spaces and surface singularities", presenting them in a more down to earth way.

Curriculum Vitae

Je n'ai pas de tâche d'enseignement en ce moment.

Liens

Séminaires

• Séminaire Géométries et Topologie, Jussieu.
• Séminaire de Géométrie Tropicale, Jussieu.
• Séminaire de Géométrie Algébrique, Jussieu.
• Séminaire sur les Singularités, Paris 7.
• RéGA, IHP.
• Géométrie et Théorie des Modèles, ENS.
• Séminaire de Géométrie Ergodique, École Polytechnique.
• SAGA, Orsay.
• Séminaires, cours et conférences à l'IHES.
• Séminaire Bourbaki, IHP.

Conférences à venir

• MathMeetings.net (tag Algebraic Geometry).
• Liste de conférences en géométrie algébrique de Ravi Vakil.
• Liste de conférences en géométrie arithmétique de Kiran Kedlaya.

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.