Classification theories and birational geometry of algebraic varieties and of their linear series. – CLASS

The classification of algebraic varieties has always been one of the main questions of algebraic geometry. The minimal model program (MMP) is one approach to classification initiated at the beginning of the '80s by the joint efforts of Kawamata, Kollár, Mori, Reid, Shokurov and many others. Its initial goal was to generalize to arbitrary dimension the birational classification of algebraic surfaces achieved by the italian school.
Work on the MMP, and particularly on finding its natural setting (that of « mildly » singular pairs) has produced a multitude of extremely powerful tools whose range of applicability goes far beyond the initial goal of classifying varieties according to the positivity of their canonical divisor.

More recently, Campana initiated another program of classification of complex projective varieties. The classification he proposes identifies two « pure » geometries: the « special » geometry and the « general type » geometry. This requires replacing varieties with particular pairs. The Campana pairs are then studied as rich geometric objects equipped with the « natural » attributes of ordinary varieties, such as morphisms, birational transformations, differential forms, fundamental group and rational points.

On the other hand, our understanding of the birational geometry of linear systems on higher dimensional algebraic varieties has greatly progressed in the last 10 years, due to the work both of the Japanese school and of Demailly, Ein, Lazarsfeld, Siu and their collaborators. It was classically known that ample linear series have beautiful geometric, cohomological and numerical properties. The introduction of new asymptotic invariants and of a powerful and flexible formalism (that of multiplier ideal sheaves) allowed them to develop a coherent theory for big divisors, whose behavior seemed mired in pathology until very recently.

There has always been a natural and close exchange of results, motivations and techniques between the classification theories and the study of the birational geometry of algebraic varieties and linear series. One such dynamic interaction culminated in the work of Hacon and McKernan: the application to the MMP of the most powerful results from the general theory of linear series and the discovery of new ones was one of the keys to Hacon and McKernan's recent breakthroughs in the field.

Our expertise, our scientific interests and the goals we intend to achieve through this project sit exactly at the crossroad of these two major research directions: the classification theories and the birational geometry of higher dimensonal varieties and of their linear series. It is an extremely fertile crossroad where recent results and techniques coming from the classification theories may deepen our understanding of linear series, and the improvement of technical tools and the development of new ones in birational geometry can potentially lead to significant advances in the classification theories. Our joint efforts will concentrate in particular on the following two main themes, which are central in the field: rational curves on singular and quasi-projective varieties, and (anti-)adjoint linear systems on higher dimensional varieties.

The moment is very favorable. Impressive progress has been accomplished and many new powerful tools are available. Nevertheless many fundamental questions remain wide open and several techniques need to be strengthened to obtain further results. This project would be a unique opportunity of solidly implanting these important research directions in France, and would permit the creation of a strong, young and complete research group, probably the first such group in continental Europe, which could interact and compete with the few existing ones, in the U.S.A., in the U.K. and in Japan.