Complex Geometric and Lie Theoretic Aspects of Hodge Theory

代数几何是研究多项式方程解的学科。这是数学的一个中心领域，有着广泛的应用，其中一小部分包括物理学、密码学和计算复杂性（更快、更有效的计算算法的设计）。霍奇理论是本学科的一个强大工具，它将线性代数结构与多项式方程系统的一组解联系起来。这种关联的优势在于，霍奇结构通常比原始的解决方案集更易于计算和分析，同时保留足够的信息以产生对解决方案集的深刻见解（以及人们想要回答的问题）。本研究项目的主要目标是发展霍奇理论，使其能够应用于代数几何中更广泛的问题。该项目将解决由霍奇理论激发的复杂几何和李理论中的一系列问题。在实践中，这意味着许多问题涉及（广义）标志流形和标志域，这些空间上的几何偏微分方程的正则系统（其积分称为水平子流形）和局部齐次空间。在Hodge理论中，标志域作为周期域出现，是极化Hodge结构的分类空间；周期域紧致对偶的标志流形几何偏微分方程的正则系统作为无穷小周期关系，Griffiths的微分方程组约束Hodge结构的变化。工作将包括：(1)将局部密尔对称空间的Satake-Baily-Borel紧化和Borel扩展定理推广到一类具有Hodge理论兴趣的局部齐次空间；(2)关于无穷小周期关系特征上同调的推测混合Hodge结构的研究；(3)启动一项计划，研究霍奇结构的变化在多大程度上取决于其特征品种。

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.