Chow Groups of Projective Varieties

Chow 射影簇群

• 批准号: 0200012
• 负责人: Chad Schoen
• 金额: $ 12万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200012&HistoricalAwards=false
• 关键词: Chow; Groups; Projective; Varieties

This project is concerned with the geometry of algebraic varieites. It builds on past experience which indicates that much can be learned about the geometry of a given algebraic variety by studying its Chow group. The Chow group is obtained from the free abelian group on all subvarieties of a given variety by modding out by the relation of rational equivalence. For algebraic curves the Chow group is a close relative of the Jacobian. While the Chow group is known to have good functorial properties, it is often very difficult to compute. Thus it is difficult to gain access to the subtle and valuable information that it frequently contains. This investigation focuses on three different aspects of the problem of computing Chow groups. First the torsion subgroup of the Chow group is being studied with an emphasis on the quotient by the subgroup generated by cycles algebraically equivalent to zero. Secondly, the relationshipbetween the Chow group and singular cohomology with rational coefficients is being investigated. The Hodge Conjecture is an important concern here. The theory of Abelian varieties and techniques from classical algebraic geometry will be brought to bear on this problem. The third line of investigation involves the relationship between the Chow group and singular cohomology with integrer coefficients. An ancient guidepost here was the so called Integral Hodge Conjecture. It has turned out that the Integral Hodge Conjecture is false in some instances. The extent of its failure is poorly understood, so the principal investigator and a PhD student are working to clarify this.To a significant extent, the value of basic research in algebraic geometry results from the tendency of mathematicians to reduce problems in various other fields of mathematics to problems in algebraic geometry. This tendency is due to the ultimate simplicity of algebraic geometry, where remarkably effective approaches to certain geometric problems have been developedwith minimal reliance on infinite processes. While the initial translation of a problem from the physical world into mathematics frequently involves limits, derivatives, integrals, and further notions involving infinite processes which go well beyond ordinary calculus, mathematicians have learned to search for hidden aspects of these problems which are essentially of an algebraic or finite nature. Neither the task of translating real world phenomena into mathematical problems, the task of discovering a hidden algebraiccore in a problem formulated using infinite processes, nor the task of solving this core problem using algebraic geometry isoften easy. Nonetheless, this lengthy process has led to profound insights. For example, algebraic geometry and some of the algebraic varieties studied by the principal investigator are of great interest to physicists in their current attempts to unify quantum mechanics with the theory of gravity.

这个项目是关于代数变量的几何学。它建立在过去的经验表明，可以学到很多关于几何的一个给定的代数品种通过研究其周组。Chow群是由一个簇的所有子簇上的自由交换群通过有理等价关系进行模化而得到的。对于代数曲线，Chow群是雅可比矩阵的近亲。虽然Chow群具有良好的函子性质，但它通常很难计算。因此，很难获得它经常包含的微妙和有价值的信息。本调查集中在三个不同方面的问题，计算周群。首先研究了Chow群的挠子群，重点研究了代数等价于零的圈所生成的子群的商。其次，研究了Chow群与有理系数奇异上同调的关系.霍奇猜想是这里的一个重要问题。从古典代数几何的阿贝尔簇和技术的理论将承担这个问题。 第三条线的调查涉及周群和奇异上同调与整系数之间的关系。一个古老的路标在这里是所谓的积分霍奇猜想。事实证明，积分霍奇猜想在某些情况下是错误的。其失败的程度知之甚少，所以首席研究员和一名博士生正在努力澄清这一点。在很大程度上，代数几何基础研究的价值源于数学家将其他数学领域的问题简化为代数几何问题的倾向。这种趋势是由于代数几何的最终简单性，其中显着有效的方法来解决某些几何问题已经开发出对无限过程的最小依赖。虽然一个问题从物理世界到数学的最初翻译经常涉及极限，导数，积分，以及涉及远远超出普通微积分的无限过程的进一步概念，但数学家已经学会了寻找这些问题的隐藏方面，这些问题本质上是代数或有限性质的。 无论是将真实的世界现象转化为数学问题的任务，还是在使用无限过程制定的问题中发现隐藏的代数核心的任务，还是使用代数几何解决这个核心问题的任务，都不容易。尽管如此，这一漫长的过程还是产生了深刻的见解。 例如，代数几何和一些主要研究者所研究的代数簇是物理学家在他们目前试图统一量子力学与引力理论时非常感兴趣的。