The theory of algebraic cycles on an arithmetical perspective.

代数几何是数学的一个分支，它将代数问题转录成几何语言。代数几何的主要研究对象是种类。多样性是由多项式方程式定义的几何对象，因此包含有关这些方程式的解的信息。因此，代数几何与数论(多项式方程的解)和拓扑学(种类的形状)密切相关。对这类变种的分类是一个传统的问题；代数曲线在19世纪由Abel和Riemann彻底研究，而代数曲面在20世纪初由意大利学派分类。现代代数几何的基础是由Grothendieck、Serre和Artin在60年代给出的。这些技术涉及上同调的概念，上同调是一种工具，它将依赖于簇形状的一些代数不变量与任何簇联系在一起。事实上，定义变化的方程式将决定变化的形状。然而，一个品种承载的信息远远不止它的形状。例如，掌握一条椭圆曲线的有理点的信息比仅仅知道它的形状要精确得多。事实上，所有的椭圆曲线都是环面的形状。作为一般原理，关于簇的算术和几何信息将给出拓扑信息，即关于簇形状的信息。我喜欢把它看作是数字和形状之间的纽带。我的目标是理解代数簇的形状是如何反过来给出关于它的几何信息的。动机的概念是由Grothendieck在60年代勾勒出来的，目的是理解域上光滑射影簇的不同上同调理论中出现的各种相似之处。Grothendieck概述了这种动机应该表现的方式，并阐明了现在所知的标准猜想。20年前，当詹森证明了半简单性猜想时，这个理论真正引起了人们的极大兴趣，他粗略地说，动机是由原子构成的。大约在那个时候，布洛赫和贝林森设想了流畅的投射变种的周氏群体将如何与他们的Grothendieck动机相关联。Bloch-Beilinson猜想是混合理据猜想范畴存在的核心，上同调是簇分类的重要工具，为簇的分类提供了拓扑不变量。Chow群构成更精细的不变量，并且具有算术和几何性质。簇的Chow群是模有理等价循环生成的自由群。虽然计算上同调群相当容易，但计算Chow群是一个具有挑战性的问题。一般而言，BB猜想规定了具有良好性质的Chow群上存在滤子，并将其与上同调环的Hodge结构联系起来。20世纪90年代末，Kimura提出了Chow动机应该表现为超向量空间而不是向量空间的想法。这个大胆的想法现在被称为木村猜想。这已经成为一个不可避免的问题，因为它意味着无效性，而且它可以在一大类品种中得到检验。已经有很多文章表明了一个品种的Chow基团的结构如何影响其上同调环的结构。我目前工作的目的是回到过去，研究上同调环的结构如何使理解Chow基团的结构成为可能。将上同调层次的性质提升到Chow群层次的关键工具是幂零猜想。最终，我的研究目的是证明木村猜想和标准猜想一起蕴含着BB猜想。我坚信，新的算术工具将是这门学科取得新突破的关键。

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.