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EP/H028870/1 - The theory of algebraic cycles on an arithmetical perspective.

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Dr C Vial EP/H028870/1 - The theory of algebraic cycles on an arithmetical perspective.

Principal Investigator - Pure Maths and Mathematical Statistics, University of Cambridge

Scheme

Postdoc Research Fellowship

Research Areas

Geometry & Topology Geometry & Topology

Start Date

04/2010

End Date

03/2013

Value

£215,562

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Grant Description

Summary and Description of the grant

Algebraic geometry is a branch of mathematics that transcribes algebraic problems into the language of geometry. The main objects of algebraic geometry are varieties. A variety is a geometric object defined by polynomial equations and thus contains information about the solutions of those equations. Therefore, algebraic geometry is closely linked to number theory (solutions to polynomial equations) and to topology (the shape of the variety). Classifying such varieties is a traditional problem; algebraic curves were thoroughly studied by Abel and Riemann in the nineteenth century, and algebraic surfaces were classified by the Italian school at the beginning of the twentieth century. The foundations of modern algebraic geometry were given by Grothendieck, Serre and Artin in the 60s. The techniques involve the notion of cohomology, which is a tool associating to any variety some algebraic invariants that depend on the shape of the variety. It is a fact that the equations defining a variety will determine the shape of the variety. However, a variety carries much more information than simply its shape. For instance, having information on the rational points of an elliptic curve is far more precise than solely knowing its shape. Indeed, all elliptic curves have the shape of a torus. As a general principle, arithmetical and geometric information about a variety will give topological information, that is information about the shape of the variety. I like to think about it as a link between the number and the shape. My ambition is to understand how, reciprocally, the shape of an algebraic variety can give geometric information about it.The concept of motive was sketched in the 60s by Grothendieck in an attempt to understand the various similarities appearing within the different cohomology theories for smooth projective varieties over a field. Grothendieck outlined the way such motives should behave and formulated what is known now as the standard conjectures. The theory really became of major interest 20 years ago when Jannsen proved the semi-simplicity conjecture, roughly stating that the motives are built out of atoms . Around that time, Bloch and Beilinson envisioned how the Chow groups of smooth projective varieties would relate to their Grothendieck motives. The Bloch-Beilinson conjectures are now at the heart of the existence of the conjectural category of mixed motives.Cohomology is an important tool in the classification of varieties and provides topological invariants for them. Chow groups constitute finer invariants and are of arithmetical and geometric nature. The Chow group of a variety is the free group generated by cycles modulo rational equivalence. While computing cohomology groups is fairly easy, computing Chow groups is a challenging problem. In general, the BB conjectures stipulate the existence of a filtration on Chow groups having nice properties and relating them to the Hodge structure of their cohomology ring.In the late 90s, Kimura came up with the idea that Chow motives should behave like super vector spaces rather than vector spaces. This audacious idea is now referred to as the Kimura conjecture. It has become an unavoidable question because of the nilpotency property it implies and the fact that it can be checked for a large class of varieties. Much has been written that shows how the structure of the Chow groups of a variety has an impact on the structure of its cohomology ring. The aim of my current work is to go backwards and study how the structure of the cohomology ring enables to understand the structure of the Chow groups. The key tool to lift properties at the level of the cohomology up to the level of Chow groups is the nilpotence conjecture. Ultimately, the aim of my research is to prove that the Kimura conjecture together with the standard conjectures implies the BB conjectures. I strongly believe that new arithmetical tools will be the key to new breakthroughs in the subject.

Structured Data / Microdata


Grant Event Details:
Name: The theory of algebraic cycles on an arithmetical perspective. - EP/H028870/1
Start Date: 2010-04-01T00:00:00+00:00
End Date: 2013-03-31T00:00:00+00:00

Organization: University of Cambridge

Description: Algebraic geometry is a branch of mathematics that transcribes algebraic problems into the language of geometry. The main objects of algebraic geometry are varieties. A variety is a geometric object defined by polynomial equations and thus contains informa ...