Hodge theory, Motives and Vanishing

本提案的主要问题是为霍奇理论找到一个可以应用于代数变族的一般框架。所提出的框架是一种带有参数的动力轮理论。特别是，会有一个关于动机局部系统的理论。后者将导致动机基本群的概念，这将提供拓扑和变种的算术之间的联系。这也与传统意义上的霍奇理论有关，比如对霍奇结构变化的研究。一个子项目涉及使用代数方法改进Kodaira消失定理。霍奇理论是这一提议的核心。它是代数几何（研究代数方程组的代数变体或解集）的一部分，与微分几何、拓扑学以及较小程度上的数学物理相互作用最为密切。数论也有一些重要的和令人惊讶的联系；动机理论也是这个提议的核心，它的出现部分是为了解释其中的一些。

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.