Prof. Dr. Carlo Gasbarri
External Senior Fellow
Oktober 2013 - September 2015
Carlo Gasbarri had 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. He has been Post-Doc 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 in many universities around the world where he gave lectures and talks at different levels.
- Paper: (with G. Pacienza et E. Rousseau), Higher dimensional tautological inequalities and applications. Math. Ann. 356 (2013), no. 2, 703-735.
- Paper: (with P. Autissier et A. Chambert--Loir), On the canonical degrees of curves in varieties of general type. Geom. Funct. Anal. 22 (2012), no. 5, 1051-1061.
- Paper: Horizontal section of connections on curves and transcendence. Acta Arith. 158 (2013), no. 2, 99–128.
- Paper: The strong $abc$ conjecture over function fields (after McQuillan and Yamanoi). Bourbaki. Vol. 2007/2008. No. 326 (2009), Exp. No. 989, viii, 219-256 (2010).
- Editor (with P. Corvaja) of the Proceedings: Arithmetic Geometry, Proceedings of the C.I.M.E. Summer School held in Cetraro (I) (2007). Springer Lecture Notes in Mathematics 2009. Contributors: J. L. Colliot, Sir P. Swinnerton Dyer, P. Vojta.
Rational Points, Rational Curves and Automorphisms of Special Varieties
The understanding of the analytic and arithmetic structures of an algebraic variety is a central problem in both algebraic/arithmetic geometry and holomorphic geometry, the latter dating at least from the classical Picard-type theorems. Currently, inroads are being made into one of the most difficult aspects of these problems, namely an understanding of the key role involved in the geometry of the canonical bundle (and related tensorial bundles) and its role in the behavior of algebraic curves (in particular rational curves) and that of rational and algebraic points. This is at least in part and in no small measure brought about by some bold structural conjectures of Serge Lang concerning algebraic varieties by which he attempted to gage analytic, algebraic and arithmetic geometry that he made explicit in the eighties. The conjectures were later formulated in a more general setting by works of Campana and a key player in formulating many effective versions of these conjectures is Vojta. By relating the behavior of rational points and similarly that of algebraic/rational curves with the geometry of the canonical bundle, Lang and Vojta's philosophy spurred many advances in our understanding of the
(Diophantine) holomorphic and arithmetic structure of algebraic varieties with many ground-breaking contributions.
The problems the project would like to attack are:
- Geometric height inequalities of Vojta for subvarieties of abelian varieties
- Contractions and classifications of two dimensional orbifolds
- Rational curves on Hyperkahler manifolds;
- affine threefolds via birational geometry.