Carlo Gasbarri

Fellowship 2013

ARCHIVE

Carlo Gasbarri is Full Professor in Mathematics at the University of Strasbourg, at the Institut de Recherche Mathématique Avancée (IRMA). He obtained his PhD in Orsay under the direction of L. Szpiro. He works in arithmetic geometry over local and global fields. More specifically he is interested in the structure of rational, integral and algebraic points on algebraic varieties and their interactions with complex analytic geometry, transcendence theory, rigid analytic geometry etc. He worked in Arakelov geometry and its applications in diophantine approximation and transcendence theory: He gave a new, more geometric, proof of Siegel theorem on integral points of hyperbolic curves which is self contained and does not rely on other big theorems as Mordell Weil theorem, Roth theorem or Schmidt subspace theorem. He also worked in the interaction within Nevanlinna theory and transcendence theory and generalized the Bombieri Schneider Lang theorem to maps from an affine variety to projective varieties. His recent interests are toward the diophantine geometry of varieties defined over function fields.

Professor Gasbarri has held post-doctiral positions in Rennes, Oxford, Kohl, Zurich, Rome. He has been Professor for 10 years at the University of Rome Tor Vergata (I) and since 2009 he is professor at the University of Strasbourg. He has been invited professor to many universities around the world.

Rational Points, Rational Curves and Automorphisms of Special Varieties

Fellows Fribourg-Strasbourg: Carlo Gasbarri, Stefan Kebekus and Gianluca Pacienza
Post-docs: Lionel Darondeau, Sergei Kovalenko and Tommaso Matteini

The attraction of mathematicians towards Diophantine equations dates as long as the Greeks but renewed in the xxth century with the geometrization of this subject. In the case of curves (which, roughly speaking corresponds to diophantine equations in two variables), a geometric invariant, the Euler characteristic, describes qualitatively the solutions in the ring of integers of a number fields (i. e. the set of integral numbers):

• when it is negative, solutions are ultimately dense and come in abundance
• solutions are still ultimately dense when the Euler characteristic is zero, but they are much sparser (logarithmic growth);
• on the opposite, there are only finitely many solutions when the Euler characteristic is positive.

The Euler characteristic also discriminates the analysis of these curves: it explains for the existence of holomorphic functions from the whole complex plane, or of meromorphic function from the Riemann sphere, valued in the given Riemann surface.
All of these questions concerning curves reveal profound interactions between arithmetics, geometry and analysis of curves and have been fully understood in the xixth and xxth centuries (theorems of Mordell–Weil, Siegel, Faltings, theorems of Picard, Nevanlinna theory).
Problems in higher dimensions are the subject of intense activity. The results are much more sparse, but there is a reasonably well-established philosophy : the conjectures of Lang–Vojta, Campana, and Green–Griffiths. However, the picture still remains largely conjectural; for example, despite important partial results, none of the cases of the above trichotomy is fully understood. In the actual state of knowledge, most geometers have split their forces for a long time: some study arithmetic, other the geometry of rational curves, other the geometry and the classification of varieties.
This project aspires to be the meeting point of some mathematicians all working in algebraic geometry, but whose interests and expertise cover all of these three aspects, united in the same philosophy and ultimately similar technical needs.

The main themes of the project are:

• The arithmetic of higher dimensional projective varieties over function fields.
• The geometry of rational curves in higher dimensional varieties and its relationship with classification. 
• The geometry of the affine varieties and its relationship with classification problems.