Chow Groups of Projective Varieties

这个项目是关于代数变分的几何学。它建立在过去的经验基础上，这些经验表明，通过研究给定代数簇的Chow群，可以学到很多关于该代数簇的几何知识。Chow群是由给定变种的所有亚种上的自由阿贝尔群通过有理等价关系建模得到的。对于代数曲线，Chow群是雅可比的近亲。虽然众所周知，Chow群具有良好的函数性，但它往往很难计算。因此，很难获得它经常包含的微妙和有价值的信息。这项研究集中在计算Chow群问题的三个不同方面。首先研究了Chow群的扭子群，重点研究了由代数等价于零的圈生成的子群的商。其次，研究了Chow群与具有有理系数的奇异上同调之间的关系。霍奇猜想在这里是一个重要的问题。经典代数几何中的阿贝尔簇理论和技巧将应用于这个问题。第三条线涉及Chow群与具有整系数的奇异上同调之间的关系。这里的一个古老路标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。其失败的程度尚不清楚，因此首席研究人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归结为代数几何问题的倾向。这种趋势是由于代数几何的极端简单性，在那里已经发展出了非常有效的方法来解决某些几何问题，而对无限过程的依赖最少。虽然一个问题从物理世界到数学的最初转换经常涉及极限、导数、积分以及涉及无限过程的进一步概念，这些概念远远超出了普通微积分的范围，但数学家已经学会了寻找这些问题的隐藏方面，这些方面本质上是代数或有限性质的。无论是将现实世界现象转化为数学问题的任务，还是在使用无限过程的问题中发现隐藏的代数核心的任务，或者使用代数几何解决这个核心问题的任务，通常都不容易。尽管如此，这一漫长的过程还是带来了深刻的见解。例如，主要研究人员研究的代数几何和一些代数变体对物理学家目前试图将量子力学与引力理论统一的努力非常感兴趣。

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.