Hodge Theory and Motives

该项目的主旨是在复数域或子域上定义的代数簇上构建相对动机理论，或者更准确地说，动机局部系统。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(它应该是阿贝尔和唐纳克的)。此外，它们应该可以通过经典拓扑和其他拓扑上的局部恒定层以及混合Hodge结构的变化来实现。希望这一理论能为家庭研究霍奇理论提供一个很好的框架。这样的理论将有许多后果：它将导致一个代数簇的动机基本群，它将与通常的基本群相关。它还可以用来研究曲线上的丛。有几个独立于上述项目的子项目。代数簇是数学中的基本对象，它们是代数方程组的解的集合。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，也就是这个项目的主题，在一定程度上受到了物理思想的推动。这个项目的目标是通过用称为动机的更简单的对象来代替霍奇变种理论来加深对它们的理解。动机的集合将足够好地反映代数变体的大部分原始结构，但它们将具有使它们更容易处理的“线性”结构。

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.