Complex Geometric and Lie Theoretic Aspects of Hodge Theory

代数几何是研究多项式方程的解的学科。这是数学的一个中心领域，有着广泛的应用，其中的一小部分包括物理、密码学和计算复杂性(设计更快、更高效的计算算法)。这门学科的一个强大工具是霍奇理论，它将线性代数结构与多项式方程组的一组解联系起来。这种关联的优势在于，Hodge结构通常比原始的解集合更易于计算和分析，同时保留了足够的信息，以产生对解集合(以及人们想要回答的关于它们的问题)的深入见解。这个研究项目的主要目标是发展霍奇理论，以便它可以应用于更广泛的代数几何问题。该项目将解决复杂几何和谎言理论中的一系列问题，这些问题是由霍奇理论推动的。在实践中，这意味着许多问题涉及(广义)旗形和旗域，这些空间上的几何偏微分方程组(其积分称为水平子流形)，以及局部齐次空间。在Hodge理论中，旗域作为周期域出现，极化Hodge结构的分类空间；标志流形作为周期域的紧对偶；几何偏微分方程组的正则系统作为无穷小周期关系；Griffiths微分方程组约束Hodge结构的变化。这项工作将包括：(1)将局部厄米特对称空间的Satake-Baly-Borel紧化和Borel扩张定理推广到一类具有Hodge理论意义的局部齐次空间；(2)研究一种猜想的混合Hodge结构关于无穷小周期关系的特征上同调；(3)启动一个程序来研究Hodge结构的变化在多大程度上由其特征簇决定。

Algebraic geometry is the study of solutions of polynomial equations. This is a central area of mathematics with a vast range of applications, a small sampling of which includes physics, cryptography, and computational complexity (the design of faster, more efficient computational algorithms). A powerful tool in the subject is Hodge theory, which associates a linear algebraic structure to a set of solutions of a system of polynomial equations. The advantage of this association is that the Hodge structure is often more amenable to computation and analysis than the original set of solutions, while at the same time retaining enough information to yield deep insights into the set of solutions (and the questions that one would like to answer about them). The principal objective of this research project is to develop Hodge theory so that it can be applied to a greater range of problems in algebraic geometry. The project will address a collection of problems in complex geometry and Lie theory that are motivated by Hodge theory. In practice this means that many of the questions concern (generalized) flag manifolds and flag domains, a canonical system of geometric partial differential equations on these spaces (whose integrals are called horizontal submanifolds), and locally homogeneous spaces. In Hodge theory, flag domains arise as period domains, the classifying spaces for polarized Hodge structures; flag manifolds as compact duals of period domains; the canonical system of geometric PDE as the infinitesimal period relation, Griffiths' system of differential equations constraining variations of Hodge structure. The work will include: (1) a generalization of the Satake-Baily-Borel compactification and Borel's extension theorem for locally Hermitian symmetric spaces to a class of locally homogeneous spaces of Hodge theoretic interest; (2) investigation of a conjectural mixed Hodge structure on the characteristic cohomology of the infinitesimal period relation; and (3) initiation of a program to study the extent to which a variation of Hodge structure is determined by its characteristic varieties.