Chow Groups of Smooth Projective Varieties

这个研究项目的重点是代数变量，即由多项式方程系统的解集定义的几何对象。品种将通过它们的Chow组进行研究。一个品种的Chow群是子品种上模于理性等价于零的元素的子群的自由阿贝尔群。虽然已知Chow族具有良好的抽象性质，但对大多数品种的结构知之甚少。该项目的目标是朝着更好地理解这种结构的各个方面取得进展。本文将研究周氏群的扭转，以及周氏群与各种上同调群之间的关系。在此基础上，利用代数等价关系研究了Chow群的商。当变量在有限域中定义时所发生的特殊性质也将被研究。该项目的调查路线延续了19世纪的历史调查。该研究将有助于更好地理解由多项式方程组的解集所定义的几何图形。它还受到大量猜想的推动，有些是最近的，有些是相当古老的，这些猜想预测了几何和数论之间美丽而微妙的关系。

This research project focuses on algebraic varieties, the geometric objects defined by the solution sets of systems of polynomial equations. Varieties will be studied through their Chow groups. The Chow group of a variety is the free abelian group on subvarieties modulo the subgroup of elements which are rationally equivalent to zero. Although the Chow group is known to have good abstract, functorial properties little is known about its structure for most varieties. The goal of the project is to make progress towards a better understanding of various aspects of this structure. The torsion in the Chow group will be studied as will the relationship between the Chow group and various cohomology groups. Furthermore, the quotient of the Chow group by the relation of algebraic equivalence will be investigated. Special properties which occur when the variety is defined over a finite field will also be studied.The line of investigation pursued in this project continues historical investigations with roots in the 19-th century. The research should contribute to a better understanding of the geometric figures defined by the solution sets to systems of polynomial equations. It is also motivated by a large body of conjectures, some recent and some quite old, which predict a beautiful and subtle relationship between geometry and the theory of numbers.