Geometry of Algebraic Cocycles

代数余循环的几何

• 批准号: 9803141
• 负责人: Jack Morava
• 金额: $ 6.51万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803141&HistoricalAwards=false
• 关键词: Geometry; Algebraic; Cocycles

9803141Gajer This project lies on the interface of algebraic topology, differentialgeometry, and algebraic geometry. It is based on the investigator'swork, in which he successfully employed some classical constructionsof algebraic topology and differential geometry to reveal thegeometric content of ordinary cohomology as well as of smooth andholomorphic Deligne cohomology. The investigator plans to apply theideas contained in his previous research on Deligne cohomology tostudy the structure of the image and the kernel of the cyclehomomorphism. Some of the deepest and most important problems oftranscendental algebraic geometry concern the cycle homomorphism. Themost celebrated among these is the Hodge conjecture. Except for a fewclasses of very special algebraic varieties, codimension one is theonly codimension in which complete cohomological and geometricdescriptions of the image and the kernel of the cycle homomorphismexist. The investigator intends to pursue extensions to allcodimensions of structures playing a central role in the codimensionone case. In particular, he introduces generalized exponentialsequences and a new cohomology theory that is conjecturally dual toChow groups of algebraic cycles modulo rational equivalence. Theobjective of this project is to verify that conjecture. If this isthe case, then the cohomology long exact sequence of the generalizedexponential sequence will induce a complete cohomological descriptionof the image and the kernel of the cycle homomorphism. Projective varieties are geometric objects defined by means ofpolynomial equations. Like other notions that arise in many importantcontexts and sound deceptively simple, for example, the ordinary wholenumbers, they possess properties that are not at all obvious or easyto discover, and over the course of many years, mathematicians havebeen led to develop elaborate algebraic machinery for makingcomputations concerning some of these properties. It is hard toremain in touch with the geometric meaning of these computations, andthis is what has given rise to the current project. The investigatorwill attempt to understand a geometric structure of what are known asalgebraic cycles. This will be done by extending some classicaltechniques of algebraic topology and differential geometry to thecontext of algebraic geometry. One might say that at least this onecorner of algebraic geometry will be reintroduced to its roots. Thiswill be not only an esthetic triumph, but an aid to intuition and thusto further discoveries and advances in mathematics and in thedisciplines like theoretical physics that make use of this mathematics.***

小行星9803141 这个项目是基于代数拓扑学、微分几何学和代数几何学的接口。 它是基于研究者的工作，他成功地利用代数拓扑和微分几何的一些经典结构，揭示了普通上同调以及光滑和全纯Deligne上同调的几何内容。 研究者计划应用他以前关于Deligne上同调的研究中所包含的思想来研究图像的结构和圈同态的核。 超越代数几何中一些最深刻和最重要的问题涉及到圈同态。 其中最著名的是霍奇猜想。 除了少数几类非常特殊的代数簇外，余维数1是唯一一个对圈同态的象和核有完备的上同调和几何描述的余维数。 研究人员打算寻求扩展到所有余维的结构中发挥核心作用的余维的情况下。 特别是，他介绍了广义exponentialsequences和一个新的上同调理论，这是一个双周群代数循环模有理等价。 这个项目的目的就是验证这个猜想。 如果是这样的话，那么广义指数序列的上同调长正合列将导出循环同态的象与核的完全上同调映射。 射影簇是用多项式方程定义的几何对象。 像其他概念一样，这些概念出现在许多重要的背景下，听起来似乎很简单，例如，普通的整数，它们拥有根本不明显或不容易发现的属性，多年来，数学家们一直在开发复杂的代数机器，用于计算其中的一些属性。 很难与这些计算的几何意义保持联系，这就是当前项目的起因。 本书将试图理解代数圈的几何结构。 这将通过扩展代数拓扑和微分几何的一些经典技巧到代数几何的背景下来完成。 有人可能会说，至少这一个角落的代数几何将重新介绍其根源。 这将不仅是一个美学上的胜利，而且有助于直觉，从而促进数学和理论物理学等学科的进一步发现和进步。