Lorenzo Fantini

Institut Mathématique de Jussieu

EN FR

About me

Lorenzo

I am a postdoc of the ANR DEFIGEO, member of the group Analyse Algébrique of the Institut Mathématique de Jussieu, in Paris.

Address

Institut Mathématique de Jussieu
Université Pierre et Marie Curie
4, place Jussieu
75252 Paris Cedex 05 - France

Office 1525-5-01 (fifth floor, between the towers 15 and 25)

Email: lorenzo.fantini@imj-prg.fr

My OpenPGP public key.

Research

Moebius's Tree
A Berkovich curve drawn by Mœbius.
I'm interested in algebraic geometry, and more precisely in non-archimedean geometry, singularity theory, tropical geometry and their interactions.
In particular, I am very interested in the connections between the singularities of algebraic varieties and the geometry of valuation spaces (especially from the point of view of Berkovich theory).
Moebius's Tree
A Berkovich curve drawn by Mœbius.

Publications and preprints

  1. "Galois descent of semi-affinoid spaces" (with D. Turchetti)
        – submitted, 27 pages (arXiv:1703.03698)
    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 XKL 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.
  2. "Normalized Berkovich spaces and surface singularities"
        – to appear in Transactions of the American Mathematical Society, doi:10.1090/tran/7209, 51 pages (arXiv)
    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.
  3. "Faithful realizability of tropical curves" (with M. Cheung, J. Park, and M. Ulirsch)
        – International Mathematics Research Notices, 2016(15), 4706–4727, 2016 (arXiv)
    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.
  4. "Normalized non-archimedean links and surface singularities"
        – Comptes Rendus Mathematique, 352(9), 719–723, 2014 (Pdf)
    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

2016-today: Postdoctoral fellow

Pierre et Marie Curie University (France).

2014-2016: Postdoctoral fellow

École Polytechnique (France), in the group of Charles Favre.

2010-2014: Doctoral fellow

University of Leuven (Belgium).

My PhD advisor was Johannes Nicaise.

2008-2010: Erasmus Mundus Master ALGANT.

I spent the first year of my Master degree at the University of Padova (Italy) and the second one at the University of Paris-Sud, Orsay (France).

I currently have no teaching duties.

If you want to find out more about me, download a complete CV (also available in French).

Links

I'm collecting some links mainly for my convenience. I share them here in case they might be useful to someone else.

Seminars

Here are some interesting seminars taking place in or around Paris:

Upcoming events

And here are some upcoming events that caught my eye. I will most likely attend only a few of them:
Lists of conferences:
Some past events I attended are archived here.