Hodge theory, Motives and Vanishing

这一建议的主要问题是为Hodge理论找到一个通用的框架，该框架可以应用于代数族。所提出的框架是一种关于动机或带有参数的动机的理论。特别是，将会有一种关于动机本地系统的理论。后者将导致基元基本群的概念，它将在变种的拓扑和算术之间提供联系。这也在更传统的意义上与霍奇理论联系在一起，就像在研究霍奇结构的变化一样。一个子项目涉及用代数方法改进Kodaira消失定理。霍奇理论是这项提议的核心。它是代数几何(研究代数方程系统的代数变体或解的集合)的一部分，它与微分几何、拓扑学和数学物理(在较小程度上)有最密切的互动。数论也有一些不平凡和令人惊讶的联系；动机理论也是这一提议的核心，它的出现在一定程度上解释了其中的一些。

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.