The theory of algebraic cycles on an arithmetical perspective.

Lead Research Organisation: University of Cambridge
Department Name: Pure Maths and Mathematical Statistics

Abstract

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.

Planned Impact

My research does not have any direct economical or industrial applications. Nonetheless, my research will benefit the competitiveness of the UK in two ways. Prior to coming to Cambridge, I was educated at the Ecole Normale Suprieure in Paris. This institution has formed numerous Nobel prize recipients and Fields medalists. I have greatly enjoyed my time in the UK and always encourage people to apply for positions there. I am also an example of a student from the ENS starting a career in the UK and my case is often referred to in careers meetings. As such, my presence in a British institution may attract promising researchers to the UK. Individual achievements in research are only the conclusion of a whole set of connections and interactions. The group dynamic is essential: a researcher making a great discovery wouldn't have made such a discovery if he hadn't been taught at University, if he hadn't interacted with other researchers, if he hadn't supervised promising students. Although he is the one making the discovery, his PhD students, the researcher next door have contributed in some way to him getting to that great idea. Having active and fruitful researchers around can only benefit the University achieving its goals of intellectual excellence. As part of the University of Cambridge, I believe I can contribute to the UK's competitiveness by confirming the University's leading role in the mathematical sphere. Moreover, I will have a more direct impact on the scientific dynamics by interacting, either through supervisions or discussions, with physicists and engineers who will certainly produce a direct contribution to the country's industrial and economical performance.

Publications

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Dao H (2015) Non-commutative resolutions and Grothendieck groups in Journal of Noncommutative Geometry

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Vial C (2010) Pure motives with representable Chow groups in Comptes Rendus Mathematique

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Vial C (2013) Niveau and coniveau filtrations on cohomology groups and Chow groups in Proceedings of the London Mathematical Society

 
Description The basic objects of algebraic geometry are called varieties. These are geometric objects defined by algebraic, that is polynomial, equations. One of the major goals of algebraic geometry is to classify varieties, or at least to define invariants that describe in a sufficiently satisfactory way some of the properties of varieties. Some invariants are of a geometric nature, while some others are of a topological or arithmetic nature. In some sense, algebraic cycles constitute some of the finest invariants attached to (smooth) varieties. Algebraic cycles are defined geometrically but also encode arithmetic and topological properties of varieties. There is a vast conjectural framework, due to Bloch and Beilinson, that links the topological properties of varieties to their geometric properties. Somewhat more specifically, Bloch and Beilinson have conjectured deep links between the Weil cohomology groups of varieties and the Chow groups of algebraic cycles of varieties.



This project has brought new insight to the framework of Bloch and Beilinson by making more precise how their conjectures are linked to Kimura's notion of finite-dimensionality. The methods deployed have made possible to prove some of the folklore conjectures (Bloch's conjecture, Kimura's conjecture, Murre's conjecture) on algebraic cycles in some new cases, thereby giving evidence that the philosophy of Bloch and Beilinson is the one to pursue.



Finally, on the one hand, collaborative work with Dao, Iyama and Takahashi has uncovered the importance of algebraic cycles in the field of non-commutative algebra. There algebraic cycles are used in an essential way to show that the existence of a so-called non-commutative resolution for a local ring is linked to the existence of a rational singularity. On the other hand, collaborative work with M. Shen has made it possible to show that algebraic cycles on certain hyperkahler fourfolds behave very much like algebraic cycles on abelian varieties. This work opens up new research directions: it is expected that the Fourier transform for algebraic cycles on abelian varieties carries through to a satisfactory theory of Fourier transform for algebraic cycles on hyperkahler varieties.

Recently, in collaboration with J. Achter (Colorado State University) and S. Casalaina-Martin (University of Colorado), we have tackled and successfully solved an arithmetic problem linked to intermediate Jacobians.
Exploitation Route My research on motives and algebraic cycles of hyperKaehler varieties has uncovered new geometric structures on hyperKaehler varieties. This will certainly spark new directions of research.
Sectors Other

 
Description My findings in pure mathematics have been used by academics at an international level to further our understanding of algebraic cycles and motives.
First Year Of Impact 2013
Sector Other
 
Description EPSRC Early Career Fellowship
Amount £401,912 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 04/2013 
End 03/2018
 
Description Chow groups of certain hyperkaehler varieties 
Organisation University of Cambridge
Country United Kingdom 
Sector Academic/University 
PI Contribution Mingmin Shen and I have had an ongoing collaboration the aim of which is to develop a theory of Fourier transform for hyperkaehler varieties which is very similar to the one developed 30 years ago by Mukai and Beauville for abelian varieties. Ongoing collaboration with Mingmin Shen, a Simons Postdoctoral Fellow at the University of Cambridge. We wrote a paper together which has been submitted for publication.
Start Year 2010
 
Description Non-commutative resolutions and Grothendieck groups 
Organisation University of Kansas
Country United States 
Sector Academic/University 
PI Contribution Dao, Iyama, Takahashi and I have been able to relate a new exciting area of noncommutative algebra concerned with the so-called noncommutative resolutions, to a more traditional area of algebraic geometry concerned with singularities of algebraic varieties, via the introduction of techniques from K-theory and algebraic cycles. This materialised in a preprint published on the arXiv in May 2012. Collaboration with Hailong Dao at the University of Kansas, and with Osamu Iyama and Ryo Takahashi at Nagoya University. I visited Hailong Dao at the University of Kansas for 3 weeks in September 2012.
Start Year 2011