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

该项目将发展霍奇理论并将其应用于代数几何、数论和表示理论中的问题。研究人员将重点关注四个相关主题：(1)Mumford-Tate （MT）域，(2)模空间，(3)代数循环和Hodge猜想，以及(4)混合Hodge模块。(1) MT域是Hodge结构的分类空间，大致来说，Mumford-Tate域的边界分量参数化了Hodge结构的退化。pi打算通过研究Mumford-Tate域及其边界分量来推进数论、表示理论和代数几何。例如，pi计划扩展Carayol的工作，Carayol试图将伽罗瓦表示与自同构表示联系起来，自同构表示的阿基量分量是离散级数的退化极限。(2)第二个主题是将几何对象的模空间实现为离散群的商。这种实现的一个例子是非超椭圆3属曲线的模空间，它可以实现为球商，其中所讨论的6维球位于K3曲面的MT域中。然而，这种类型的例子并不多。pi打算寻求更多。(3)第三个主题涉及到通过格林和格里菲斯的正规函数及其奇异性来解决霍奇猜想的方法。pi将在几个方向上发展这种方法。例如，他们将研究与正规函数相关的阿基米德高度函数，他们打算研究与正规函数相关的非约MT群。(4)最后，pi将开发混合Hodge模块复杂变化的灵活理论，并将其应用于表征理论中出现的问题。特别是，他们想要理解在稳定曲线的模空间上作为复杂混合Hodge模的共形块的结构。霍奇理论是代数几何的一个中心领域，其根源在于经典（19世纪）的特殊函数和周期积分理论。从现代的观点来看，霍奇理论的目标是将代数变量的拓扑不变量与算术不变量和解析不变量联系起来。其中心思想是代数变体上同调群上的霍奇结构。而上同群是纯拓扑的，只取决于变化的形状，霍奇结构是一个更敏感的不变量。因此，霍奇结构携带了大量重要的代数几何和数论信息。在代数几何中最著名的未解问题是Hodge猜想，它是关于一簇上同调群的Hodge结构与某些子簇的存在性之间的关系的问题。这种对拓扑对象与精细解析不变量之间关系的关注是整个霍奇理论的典型特征，也是该FRG所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，包括数论、代数几何和表示理论。由于涉及的技术数量众多，pi拥有各种各样的技能和观点。FRG的一个重要组成部分将专门用于会议，这些会议将在pi之间交流思想，并在与霍奇理论有关的广泛主题上培训博士后研究员和研究生。

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.