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Cohomological study of algebraic varieties – ECOVA

ECOVA

Cohomological study of algebraic varieties

Purposes

This project is at the crossroads of algebraic geometry and number theory. It aims at investigating a series of problems arising as consequences or particular cases of `conjectures horizon' (such as the standard conjectures, the Hodge and the Tate conjectures, various conjectures on algebraic cycles), which can be roughly classified according to the three following main directions:<br /><br /> Complex and l-adic coefficients: absolute setting (special cases of the Hodge and the Tate conjectures, Mumford-Tate conjecture), variational setting (geometric study of the exceptional degeneracy loci, representations of the fundamental group, l-independence, Shimura varieties);<br /> Integral, finite, adelic coefficients: absolute setting (obstructions to the integral variant of the Hodge and the Tate conjectures), variational setting (representations of the fundamental group, semi simplicity modulo l, adelic openness and Shimura varieties);<br /> Finer questions on algebraic cycles (Higher Abel-Jacobi maps, integral and birational aspects, study of zero-cycles and rational points, mixed motives).

The 60 months of the project will be structured around three conference-like events (Opening, mid-term and closing meetings) but the main features of ECOVA is to create and maintain constant interactions between the members of the project and the members of the project and outside researchers by a series of more specific regular events (working groups, one-day meetings, invited lectures, seminars) organized all year round as well as by supporting visit to and invitation of outside members, attending conferences. This organisation should provide a constant and uniform progress in the five-years period of our project.

On the period 01/10/2015-31/03/2017, the scientific production of the members of the project is outstanding as shown by the many articles already written and the international outreach. The interaction between the memebers was constant, which was made possible through the two monthly seminars they coorganize (‘Autour des Cycles algébriques’ and ‘Variétés rationnelles’, where several of the non-parisian members where invited), the various conferences that gathered several members of ECOVA, missions/invitations, the organization of joint projects (working groups, invited lectures, the CIRM 2018 conference etc.). Various internal collaborations strated and should give rise to joint papers.

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On the period 01/10/2015-31/03/2017, 22 preprints were written, the members gave more than a hundred talks in international conferences and in seminars - both in France and aborad.

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.

Project coordination

Anna Cadoret (Centre de Mathématiques Laurent Schwartz)

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.

Partner

CMLS Centre de Mathématiques Laurent Schwartz

Help of the ANR 166,400 euros
Beginning and duration of the scientific project: September 2015 - 48 Months

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