Special varieties

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.