Geometry of Algebraic Cocycles

这个项目涉及到代数拓扑学、微分几何和代数几何。它基于研究者的工作，他成功地利用代数拓扑和微分几何的一些经典构造来揭示普通上同调以及光滑和全纯Deligne上同调的几何内容。这位研究人员计划应用他以前对Deligne上同调的研究中所包含的思想来研究像的结构和圈同态的核。超越代数几何的一些最深和最重要的问题与圈同态有关。其中最著名的是霍奇猜想。除少数几类非常特殊的代数簇外，余维1是唯一存在像和圈同态的核的完全上同调和几何描述的余维。调查人员打算将在辅维酮案件中发挥核心作用的结构的所有协维度进行扩展。特别地，他引入了广义指数序列和一个新的上同调理论，即对模有理等价的Chow代数圈群猜想对偶。这个项目的目标是验证这一猜想。如果是这样，则广义可积序列的上同调长正合列将诱导出对像和圈同态的核的完全上同调描述。射影簇是用多项式方程定义的几何对象。就像在许多重要的上下文中出现的、听起来很简单的其他概念一样，例如普通的整数，它们具有根本不明显或不容易发现的性质，在许多年的过程中，数学家们被引导开发出复杂的代数机器来进行关于这些性质的计算。很难理解这些计算的几何意义，这就是目前这个项目的起因。研究人员将试图理解所谓的代数圈的几何结构。这将通过将代数拓扑和微分几何的一些经典技巧推广到代数几何的背景下来实现。有人可能会说，至少代数几何的这个核心将被重新引入它的根。这将不仅是美学上的胜利，而且是对直觉的帮助，有助于进一步发现和进步数学，以及利用这种数学的学科，如理论物理学。

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.***