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.

该提案中的主要问题涉及找到霍奇理论的一般框架，该框架可以应用于代数品种的家庭。提出的框架是一种具有参数的动机滑轮或动机的理论。特别是，将有一个动机本地系统的理论。然后，后者将导致动机基本群体的概念，这将提供拓扑结构和品种算术之间的联系。这在更传统的意义上也与霍奇理论联系起来，就像研究霍奇结构的变化一样。一个suproject涉及使用代数方法对kodaira消失的改进。hodge理论是该提案的核心。它是代数几何形状的一部分（代数品种或代数方程系统解决方案集的研究），它与差异几何，拓扑，拓扑以及较小程度的数学物理学最紧密地相互作用。数字理论也存在非平凡且令人惊讶的联系。动机理论也是该提议的核心，部分原因是解释其中一些。