FRG: Collaborative Research: Hodge Theory, Moduli, and Representation Theory

FRG：协作研究：霍奇理论、模数和表示理论

• 批准号: 1361147
• 负责人: Matthew Kerr
• 金额: $ 30.19万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361147&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Hodge; Theory

The project will develop Hodge theory and apply it to problems in algebraic geometry, number theory and representation theory. The researchers intend to focus on four related topics: (1) Mumford-Tate (MT) domains, (2) moduli spaces, (3) algebraic cycles and the Hodge conjecture, and (4) mixed Hodge modules. (1) MT domains are classifying spaces of Hodge structures, and, roughly speaking, the boundary components of Mumford-Tate domains parametrize degenerations of Hodge structures. The PIs intend to advance number theory, representation theory and algebraic geometry by studying Mumford-Tate domains and their boundary components. For example, the PIs plan to extend work of Carayol, which seeks to associate Galois representations to automorphic representations whose archimedian component is a degenerate limit of discrete series. (2) The second topic concerns the realization of moduli spaces of geometric objects as quotients of discrete groups. An example of such a realization is the moduli space of non-hyperelliptic genus 3 curves, which can be realized as a ball quotient, where the 6 dimensional ball in question sits in the MT domain of K3 surfaces. However, there are not many examples of this type known. The PIs intend to look for more. (3) The third topic involves the approach to the Hodge conjecture via normal functions and their singularities due to Green and Griffiths. The PIs will develop this approach in several directions. For example, they will study the archimedean height function associated to a normal function, and they intend to study the non-reductive MT groups associated to normal functions. (4) Finally, the PIs will develop a flexible theory of complex variations of mixed Hodge modules and apply it to questions arising in representation theory. In particular, they would like to understand the structure of conformal blocks viewed as complex mixed Hodge modules on the moduli spaces of stable curves.Hodge theory is a central area of algebraic geometry with roots in the the classical (19th century) theory of special functions and period integrals. From a modern point of view, the goal of Hodge theory is to relate topological invariants of algebraic varieties to arithmetic and analytic invariants. The central notion is that of a Hodge structure on the cohomology groups of an algebraic variety. While the cohomology groups are purely topological, depending only on the shape of variety, the Hodge structure is a much more sensitive invariant. Consequently, the Hodge structure carries a great deal of important algebro-geometric and number-theoretical information. The most famous unsolved problem in algebraic geometry is the Hodge conjecture, a question about the relationship between the Hodge structure of the cohomology groups of a variety and the existence of certain subvarieties. This focus on the relationship between topological objects and finer analytic invariants is typical of Hodge theory as a whole, and it is the main motivation for the research supported by this FRG. This research will consequently impact several areas of mathematics including number theory, algebraic geometry and representation theory. Owing to the number of techniques involved, the PIs have a diverse set of skills and points of view. An important component of the FRG will be devoted to conferences, which will exchange ideas between the PIs and train postdoctoral fellows and graduate students in a wide range of topics having to do with Hodge theory.

该项目将发展霍奇理论，并将其应用于代数几何，数理论和代表理论中的问题。 研究人员打算关注四个相关主题：（1）Mumford-tate（MT）域，（2）模量空间，（3）代数循环和Hodge猜想，以及（4）混合Hodge模块。 （1）MT域是分类霍奇结构的空间，并且大致说明Mumford-Tate域的边界成分参数化Hodge结构的退化。 PI打算通过研究Mumford-Tate域及其边界组成部分来推进数字理论，表示理论和代数几何形状。 例如，PIS计划扩展Carayol的工作，该工作旨在将Galois表示形式与Archimedian组件的自动形式表示相关联。 （2）第二个主题涉及将几何对象的模量空间作为离散组的商的实现。 这种实现的一个例子是非hyperelliptic属3曲线的模量空间，可以将其实现为球商，其中有6个维球位于K3表面的MT域中。 但是，这种类型的示例并不多。 PI打算寻找更多。 （3）第三个主题涉及通过正常功能及其由于绿色和格里菲斯引起的奇异性的方法。 PI将在多个方向上开发这种方法。 例如，他们将研究与正常功能相关的阿基米德高度功能，并打算研究与正常功能相关的非还原性MT组。 （4）最后，PIS将发展出混合霍奇模块的复杂变化的灵活理论，并将其应用于表示理论中引起的问题。 特别是，他们想理解在稳定曲线的模量空间上被视为复杂混合霍奇模块的结构结构。Hodge理论是代数几何学的中心区域，具有古典（19世纪）特殊功能和周期积分理论中的根源。 从现代的角度来看，霍奇理论的目标是将代数品种的拓扑不变性与算术和分析不变性联系起来。 中心概念是代数品种的同胞组上的霍奇结构的概念。 虽然共同体组纯粹是拓扑结构，但仅取决于多样性的形状，但霍奇结构是一种更敏感的不变性。因此，霍奇结构带有大量重要的代数几何和数字理论信息。 代数几何形状中最著名的未解决问题是霍奇猜想，这是一个关于多样性组的杂物结构与某些亚地区存在的问题之间的关系。 关注拓扑对象与更精细的分析不变的关系的关注是整个霍奇理论的典型代表，这是该FRG支持的研究的主要动机。 因此，这项研究将影响数学的几个领域，包括数字理论，代数几何和代表理论。 由于涉及的技术数量，PI具有多种技能和观点。 FRG的一个重要组成部分将致力于会议，该会议将在PIS和火车博士后研究员和研究生之间交换各种主题，这与Hodge理论有关。