Chow Groups of Smooth Projective Varieties

平滑射影簇的 Chow 群

• 批准号: 9970500
• 负责人: Chad Schoen
• 金额: $ 9万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970500&HistoricalAwards=false
• 关键词: Chow; Groups; Smooth; Projective; Varieties

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.

本研究项目的重点是代数簇，由多项式方程组的解集定义的几何对象。 品种将通过他们的周组进行研究。 一个簇的Chow群是子簇上的自由阿贝尔群，模是有理等价于零的元素的子群。 虽然Chow群被认为具有很好的抽象和函子性质，但对于它的大多数变种的结构知之甚少。 该项目的目标是在更好地了解这一结构的各个方面取得进展。 在周群的挠将被研究，将周群和各种上同调群之间的关系。 进一步，利用代数等价关系研究了Chow群的商. 特殊性质时出现的品种被定义在一个有限域也将被研究。在这个项目中追求的调查线继续历史调查与根在19世纪世纪。 该研究将有助于更好地理解由多项式方程组的解集定义的几何图形。 它也是由大量的插图，一些最近的和一些相当古老的，预测几何和数论之间的美丽和微妙的关系。