This project, at the crossroads of algebraic geometry and number theory, aims at investigating some aspects of the relationship between the arithmetico-geometric and cohomological properties of an algebraic variety.
The notion of cohomology is somewhat the modern culmination of the old idea of `linearization', which consists in associating to an object with a rich structure (an algebraic variety $X$), families of simpler objects (linear i.e. vector spaces or modules with additional data) supposed to encode a lot of the relevant information about the original object but easier to handle, classify etc.
Abelian cohomology has progressively emerged in topology and complex geometries during the first half of the twentieth century. During this period, first, and second cohomology pointed sets also appear with the study of twists in Galois theory. In the fifties and early sixties, the development of category theory and homological algebra provided the conceptual frame to elaborate systematically a cohomological formalism in abelian categories, paving the way to its diffusion in many areas of mathematics. In particular, it is intrinsically connected to the breathtaking development of modern algebraic geometry, yielding to the introduction of various algebraic Weil cohomologies related by comparison isomorphisms. Around the early seventies, these several cohomological avatars were unified in Grothendieck's fascinating conjectural theory of pure motives, which was later included in the wider theory of mixed motives.
These theories - partly conjectural - naturally give rise to `conjectures horizon', such as the Hodge and the Tate conjectures, various conjectures on algebraic cycles etc. The purpose of ECOVA is to study a series of problems arising as consequences or particular cases of these `conjectures horizon' and which can be roughly classified following three main directions:
- complex and l-adic coefficients: absolute setting (special cases of the Hodge and the Tate conjectures), variational setting (geometric study of the exceptional loci, representations of the fundamental group, l-independence, varietes de Shimura);
- integral, finite, adelic coefficients: absolute setting (obstructions to the Hodge and the Tate conjectures), variational setting (representations of the fundamental group, l-independence);
- Finer questions on algebraic cycles (Higher Abel-Jacobi maps, integral and birational aspects, study of zero-cycles and rational points, mixed motives).
ECOVA gathers young mathematicians working on the various aspects described above of deep long-standing problems in arithmetic geometry so that they could share the technics they handle, acquire new ones, initiate and pursue collaborations. As a result, a particular emphasis will be put on the organization of weekly or monthly working groups, series of invited lectures of post-doctoral level, visit to and invitation of foreign specialists, attending conferences.The 48 months of the project will be structured around three conference-like events: an opening meeting where the researchers involved in the project will present and discuss the mathematical problems they are working on, a mid-project meeting, in the spirit of a mixed summer school/research conference and a closing meeting, with the purpose of presenting the main mathematical achievements of the projects in the frame of a wide-audience research conference.
Centre de Mathématiques Laurent Schwartz (Laboratoire public)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
Centre de Mathématiques Laurent Schwartz
Help of the ANR 166,400 euros
Beginning and duration of the scientific project: September 2015 - 48 Months