Hodge theory, Motives and Vanishing

霍奇理论、动机和消失

• 批准号: 1201031
• 负责人: Donu Arapura
• 金额: $ 18.23万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201031&HistoricalAwards=false
• 关键词: Hodge; theory; Motives; Vanishing

The main problems in this proposal concern finding a general framework for Hodge theory, which can be applied to families of algebraic varieties. The proposed framework is a theory of motivic sheaves or motives with parameters. In particular, there would be a theory of motivic local systems. The latter would then lead to a notion of motivic fundamental group, which would provide a link between the topology and arithmetic of varieties. This also connects to Hodge theory in a more traditional sense, as in the study of variations of Hodge structures. A sub-project involves refinements of the Kodaira vanishing theorem using algebraic methods.Hodge theory lies at the core of this proposal. It is the part of algebraic geometry (the study of algebraic varieties or sets of solutions of systems of algebraic equations) that interacts most closely with differential geometry, topology and to a lesser extent, mathematical physics. There are nontrivial and surprising connections with number theory as well; the theory of motives, which is also central to this proposal, arose in part to explain some of these.

在这个建议的主要问题涉及找到一个一般框架霍奇理论，它可以适用于家庭的代数簇。建议的框架是一个理论的动机层或动机参数。特别是，会有一个动机局部系统的理论。后者将导致motivic基本群的概念，这将提供拓扑和算术之间的联系品种。这也连接到霍奇理论在更传统的意义上，如在研究的变化霍奇结构。一个子项目涉及使用代数方法改进科代拉消失定理。霍奇理论是这个建议的核心。它是代数几何的一部分（研究代数簇或代数方程组的解），与微分几何，拓扑学和数学物理学最密切地相互作用。它与数论也有着非同寻常和令人惊讶的联系;动机理论也是这一提议的核心，它的出现部分地解释了其中的一些。