# Geometry and arithmetics through the theory of algebraic cycles

Lead Research Organisation:
University of Cambridge

Department Name: Pure Maths and Mathematical Statistics

### Abstract

The basic objects of algebraic geometry are algebraic varieties. These are defined locally as the zero locus of polynomial equations. The main goal of algebraic geometry is to classify varieties. An approach consists in attaching invariants to varieties. Some invariants are of an arithmetic nature, e.g. the gcd of the degrees of closed points on X. Some are of a topological nature, e.g. the singular cohomology of the underlying topological space of X. Some are of a geometric nature, e.g. the Chow groups of X. A codimension-n algebraic cycle on X is a formal sum of irreducible subvarieties of codimension n and the Chow group CH^n(X) is the abelian group with basis the irreducible subvarieties of codimension n in X modulo a certain equivalence relation called rational equivalence. Roughly, rational equivalence is the finest equivalence relation on algebraic cycles that makes it possible to define unambiguously an intersection product on cycles. Moreover, the aforementioned invariants for X are encoded (or at least expected to be) in the Chow groups X. Therefore, in some sense, algebraic cycles are the finest invariants for algebraic varieties, and the theory of algebraic cycles lies at the very heart of geometry, topology and number theory.

I will integrate methods from K-theory, Galois cohomology and number theory to derive new results in the theory of algebraic cycles on varieties defined over finitely generated fields or other fields of arithmetic interest. Conversely, I will use the theory of algebraic cycles to derive new results of arithmetic

interest. In addition, the outcome of such results will shed new light on the geometry of such varieties. Thus, by its very nature, my research proposal on the theory of algebraic cycles is intradisciplinary within the mathematical sciences.

I will integrate methods from K-theory, Galois cohomology and number theory to derive new results in the theory of algebraic cycles on varieties defined over finitely generated fields or other fields of arithmetic interest. Conversely, I will use the theory of algebraic cycles to derive new results of arithmetic

interest. In addition, the outcome of such results will shed new light on the geometry of such varieties. Thus, by its very nature, my research proposal on the theory of algebraic cycles is intradisciplinary within the mathematical sciences.

### Planned Impact

Algebraic geometry and number theory occupy a central place in pure mathematics and have interactions with many other fields ranging from representation theory, combinatorics, Lie theory, symplectic geometry, as well as more applied fields including cryptography (including elliptic curve cryptography) and error correction codes. Therefore, advances in the fields of algebraic geometry and number theory have potential impact on everyday life technology such as CD players and the internet through error correction codes, and security encryption for credit cards, etc.

The theory of algebraic cycles brings together algebraic geometry and number theory and is rooted at the very heart of those two central and most ancient fields of mathematics. As such, the significance of algebraic cycles within mathematics cannot be underestimated.

Problems around algebraic cycles are considered to be amongst the most difficult and most challenging problems in pure mathematics. Any significant result obtained as part of this proposal will attract interest from around the world, including many parts of Europe, Russia, the United States, India, China and Japan. Any progress in this field will enhance the international reputation of the Host Institution and, consequently, help Institutions across the UK recruit top-class academics and students.

The results of this work will be disseminated through papers that will first be uploaded on both the arXiv and my homepage at the Mathematical Department's website to ensure a fast and efficient dissemination of the research in order to maximize their usefulness and impact to academia. The results of this work will then be published in leading international journals to attest the quality of the research and increase the prestige of both the EPSRC and the Host Institution.

The knowledge will also be shared through the PI's travels to conferences and seminars, as well as by inviting visitors to the Host Institution. I will emphasize the cross-disciplinary perspective of my research and aim both at a general audience and at experts in my field. I am already familiar with this exercise as I have already presented the technical details of my work at several conferences, workshops and specialists' seminar series through Europe and Japan. I am also experienced at aiming at a general audience of mathematicians and I have recently been invited to give a Colloquium talk on my

research in Liverpool. There I have especially focused on the interactions of geometry and arithmetics in the theory of algebraic cycles.

Finally, the cross-disciplinary research undertaken will be disseminated at a graduate level through the master's program in mathematics (Part III Tripos) at Cambridge by designing graduate courses and setting up essays emphasizing the ubiquity of the theory of algebraic cycles. For instance, I designed

a graduate course at Cambridge for the Part III Tripos entitled `Galois cohomology'. This was a demanding task but I thoroughly enjoyed showing the students how techniques from algebra, algebraic geometry, number theory and homological algebra can be brought together and illuminate each other.

This way I am directly responsible for developing the students' ability to identify and analyze patterns, logic and critical thinking skills, ability to see relationships, and problem solving skills. Statistically, students with a postgraduate degree in the mathematical sciences (this includes master's degrees and PhDs) have the highest average starting salary among all UK holders of postgraduate degrees, according to Adrian Smith's report One Step Beyond (March 2010), p. 94: http://www.bis.gov.uk/one-step-beyond.

Thus teaching advanced topics such as my research to master's students makes those students very desirable to employers and feeds in the critical

pathway towards economic and societal impact.

The theory of algebraic cycles brings together algebraic geometry and number theory and is rooted at the very heart of those two central and most ancient fields of mathematics. As such, the significance of algebraic cycles within mathematics cannot be underestimated.

Problems around algebraic cycles are considered to be amongst the most difficult and most challenging problems in pure mathematics. Any significant result obtained as part of this proposal will attract interest from around the world, including many parts of Europe, Russia, the United States, India, China and Japan. Any progress in this field will enhance the international reputation of the Host Institution and, consequently, help Institutions across the UK recruit top-class academics and students.

The results of this work will be disseminated through papers that will first be uploaded on both the arXiv and my homepage at the Mathematical Department's website to ensure a fast and efficient dissemination of the research in order to maximize their usefulness and impact to academia. The results of this work will then be published in leading international journals to attest the quality of the research and increase the prestige of both the EPSRC and the Host Institution.

The knowledge will also be shared through the PI's travels to conferences and seminars, as well as by inviting visitors to the Host Institution. I will emphasize the cross-disciplinary perspective of my research and aim both at a general audience and at experts in my field. I am already familiar with this exercise as I have already presented the technical details of my work at several conferences, workshops and specialists' seminar series through Europe and Japan. I am also experienced at aiming at a general audience of mathematicians and I have recently been invited to give a Colloquium talk on my

research in Liverpool. There I have especially focused on the interactions of geometry and arithmetics in the theory of algebraic cycles.

Finally, the cross-disciplinary research undertaken will be disseminated at a graduate level through the master's program in mathematics (Part III Tripos) at Cambridge by designing graduate courses and setting up essays emphasizing the ubiquity of the theory of algebraic cycles. For instance, I designed

a graduate course at Cambridge for the Part III Tripos entitled `Galois cohomology'. This was a demanding task but I thoroughly enjoyed showing the students how techniques from algebra, algebraic geometry, number theory and homological algebra can be brought together and illuminate each other.

This way I am directly responsible for developing the students' ability to identify and analyze patterns, logic and critical thinking skills, ability to see relationships, and problem solving skills. Statistically, students with a postgraduate degree in the mathematical sciences (this includes master's degrees and PhDs) have the highest average starting salary among all UK holders of postgraduate degrees, according to Adrian Smith's report One Step Beyond (March 2010), p. 94: http://www.bis.gov.uk/one-step-beyond.

Thus teaching advanced topics such as my research to master's students makes those students very desirable to employers and feeds in the critical

pathway towards economic and societal impact.

## People |
## ORCID iD |

Charles Vial (Principal Investigator / Fellow) |

### Publications

Achter J
(2017)

*On descending cohomology geometrically*in Compositio Mathematica
ACHTER J
(2018)

*Derived equivalent threefolds, algebraic representatives, and the coniveau filtration*in Mathematical Proceedings of the Cambridge Philosophical Society
Charles Vial (Author)
(2013)

*Algebraic cycles and fibrations*in Documenta Mathematica
Charles Vial (Author)
(2013)

*Projectors on the intermediate algebraic Jacobians*in New York Journal of Mathematics
Fu L
(2019)

*Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties*in Geometry & Topology
Shen M
(2016)

*The Fourier Transform for Certain HyperKähler Fourfolds*in Memoirs of the American Mathematical Society
SHEN M
(2016)

*THE MOTIVE OF THE HILBERT CUBE*in Forum of Mathematics, Sigma
Vial C
(2017)

*On the motive of some hyperKähler varieties*in Journal für die reine und angewandte Mathematik (Crelles Journal)
Vial C
(2019)

*Parameter Spaces for Algebraic Equivalence*in International Mathematics Research Notices
Vial C
(2017)

*Exceptional collections, and the Néron-Severi lattice for surfaces*in Advances in MathematicsDescription | On the one hand, I have discovered new and surprising geometric properties of hyperKaehler varieties. HyperKaehler varieties are important objects in geometry that come equipped with a very rich structure (which explains their significance). They are somehow closely related to so-called Calabi-Yau varieties, whose importance in theoretical physics and in our understanding of the universe cannot be underestimated. |

Exploitation Route | My findings have contributed to the academic knowledge in the field of algebraic geometry. |

Sectors | Other |

Description | My findings in the theory of algebraic cycles on hyperKaehler varieties have sparked new directions of research in pure mathematics. I have initiated a fruitful collaboration with Lie Fu (Lyon, France); we have now three published articles, as well as four preprints, some of them with Robert Laterveer (Strasbourg, France). Our current findings aim at unraveling the links between the derived category and the Chow motive of hyperKaehler varieties. My collaboration with Jeff Achter and Sebastian Casalaina-Martin from the University of Colorado and the Colorado State University has grown deeper: we have proved further results of arithmetic significance concerning algebraic cycles. This has materialised in five published papers, one preprint, and still on-going projects. Our results are currently being used by us and others to study rationality questions over fields of arithmetic significance (non-algebraically closed, positive characteristic). |

Sector | Other |

Impact Types | Cultural |

Description | Motives of hyperKahler varieties |

Organisation | University of Amsterdam |

Department | Mathematics Department |

Country | Netherlands |

Sector | Academic/University |

PI Contribution | We conjectured the existence motivic structures for hyperKaehler varieties that are similar to that of abelian varieties. We established such existence in important cases. |

Collaborator Contribution | We conjectured the existence motivic structures for hyperKaehler varieties that are similar to that of abelian varieties. We established such existence in important cases. |

Impact | 2 published articles. |

Start Year | 2012 |

Description | Motivic hyperKaehler resolution conjecture for generalized Kummer varieties |

Organisation | Joseph Fourier University |

Country | France |

Sector | Academic/University |

PI Contribution | We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties. |

Collaborator Contribution | We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties. |

Impact | A preprint on the arXiv that has been submitted for publication. |

Start Year | 2016 |

Description | Motivic hyperKaehler resolution conjecture for generalized Kummer varieties |

Organisation | University of Lyon |

Department | Department of Mathematics |

Country | France |

Sector | Academic/University |

PI Contribution | We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties. |

Collaborator Contribution | |

Impact | A preprint on the arXiv that has been submitted for publication. |

Start Year | 2016 |

Description | On descending cohomology geometrically |

Organisation | Colorado State University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | We wrote 3 articles, 1 of which has been accepted for publication at Compositio Math, and 2 of which have been posted on the arXiv and have been submitted for publication. |

Collaborator Contribution | Joint articles |

Impact | Series of articles. Multi-disciplinary in that the projects deals with arithmetic questions related to geometry. |

Start Year | 2015 |

Description | On descending cohomology geometrically |

Organisation | University of Colorado Boulder |

Department | Department of Mathmatics |

Country | United States |

Sector | Academic/University |

PI Contribution | We wrote 3 articles, 1 of which has been accepted for publication at Compositio Math, and 2 of which have been posted on the arXiv and have been submitted for publication. |

Collaborator Contribution | Joint articles |

Impact | Series of articles. Multi-disciplinary in that the projects deals with arithmetic questions related to geometry. |

Start Year | 2015 |