Hodge Theory and Motives

霍奇理论和动机

• 批准号: 0754127
• 负责人: Donu Arapura
• 金额: $ 14.1万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0754127&HistoricalAwards=false
• 关键词: Hodge; Theory; Motives

The main thrust of the project is the construction of a theory of relative motives, or more precisely motivic local systems, over an algebraic variety defined over a field of complex numbers or a subfield. These objects should exist in the abstract, and form a category with good properties (it should be abelian and tannakian). Furthermore they should be realizable by locally constant sheaves on the classical and etale topologies, as well as by variations of mixed Hodge structures. The hope is that this theory should provide a good framework in which to study Hodge theory for families. Such a theory would have a number of ramifications: It would lead to the motivic fundamental group of an algebraic variety which would be related to the usual fundamental group. And it could used in the study of bundles over a curve. There are a couple of subprojects which are separate from the above. These involve the study of vanishing theorems by positive characteristic techniques, and the study of Kaehler-de Rham cohomology.Algebraic varieties are basic objects in mathematics; they are sets of solutions of systems of algebraic equations. Their study has found interactions with areas as diverse as mathematical physics and cryptography. Indeed Hodge theory, which is the subject of this project, was partly motivated by physical ideas. The goal of this project is to further the understanding of the Hodge theory of varieties by replacing them with simpler objects called motives. The collection of motives would be fine enough to reflect much of the original structure of algebraic varieties, but they would posses a "linear" structure which makes them easier to handle.

该项目的主旨是建设一个理论的相对动机，或更准确地motivic当地系统，在代数品种定义的领域复杂的数字或一个子域。这些对象应该抽象地存在，并形成一个具有良好性质的范畴（它应该是阿贝尔和坦纳基的）。此外，他们应该是可实现的局部常数层的经典和etale拓扑结构，以及混合霍奇结构的变化。希望这个理论能提供一个很好的框架来研究霍奇家庭理论。 这样的理论会有许多分支：它会导致一个代数簇的motivic基本群，它会与通常的基本群相关。它可以用于研究曲线上的丛。有几个子项目与上述项目分开。这些涉及消失定理的研究，积极的特征技术，并研究Kaehler-de Rham上同调。代数品种是数学的基本对象，他们是一套解决方案的系统代数方程。 他们的研究发现了与数学物理和密码学等不同领域的相互作用。事实上，霍奇理论，这是本项目的主题，部分动机是物理思想。这个项目的目标是进一步理解霍奇理论的品种取代他们更简单的对象称为动机。 动机的集合将是足够好的，以反映大部分的原始结构的代数簇，但他们将是一个“线性”的结构，使他们更容易处理。