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The theory of algebraic cycles on an arithmetical perspective.

算术角度的代数环理论。

基本信息

批准号：
EP/H028870/1
负责人：
Charles Vial
金额：
$ 27.47万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH028870%2F1
关键词：
theory algebraic cycles arithmetical perspective.

项目摘要

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.

代数几何是将代数问题转化为几何语言的数学的一个分支。代数几何的主要对象是簇。变量是由多项式方程定义的几何对象，因此包含有关这些方程的解的信息。因此，代数几何与数论（多项式方程的解）和拓扑学（簇的形状）密切相关。分类这样的品种是一个传统的问题;代数曲线彻底研究了阿贝尔和黎曼在19世纪，和代数曲面分类的意大利学校在20世纪初的世纪。现代代数几何的基础是由格罗滕迪克、塞尔和阿廷在60年代提出的。该技术涉及上同调的概念，这是一个工具，任何品种的一些代数不变量，取决于形状的品种。定义一个品种的方程将决定品种的形状，这是一个事实。然而，一个品种携带的信息远不止它的形状。例如，拥有关于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，所有椭圆曲线都有环面的形状。作为一般原则，关于一个簇的算术和几何信息将给出拓扑信息，即关于簇的形状的信息。我喜欢把它看作是数字和形状之间的联系。我的目标是了解如何，但，形状的代数簇可以给几何信息。动机的概念是勾勒在60年代的Grothendieck在试图了解各种相似性出现在不同的上同调理论光滑投影品种在一个领域。格罗滕迪克概述了这些动机的行为方式，并制定了现在被称为标准假设的东西。这个理论在20年前真正引起了人们的兴趣，当时詹森证明了半简单性猜想，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光滑投射变体的周群如何与他们的格罗滕迪克动机联系起来。Bloch-Beilinson拓扑是混合动机的拓扑范畴存在的核心，上同调是簇分类的重要工具，为簇提供拓扑不变量。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群是由循环模有理等价生成的自由群。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。一般来说，BB结构规定了Chow群上存在一个滤子，这些滤子具有良好的性质，并将它们与它们的上同调环的Hodge结构联系起来。在90年代后期，Kimura提出了Chow motives应该表现得像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。它已经成为一个不可避免的问题，因为它意味着幂零性和事实，它可以检查的一大类品种。很多已经写的，表明如何结构的周群的品种有影响的结构，其上同调环。我目前的工作的目的是回去和研究如何上同调环的结构，使了解结构的周群。将上同调水平上的性质提升到Chow群水平的关键工具是零猜想。最终，我研究的目的是证明木村猜想与标准猜想一起隐含BB猜想。我坚信，新的算术工具将是这一学科取得新突破的关键。