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、数论和表示论中的问题。研究人员打算集中于四个相关的主题：(1)Mumford-Tate(MT)域，(2)模空间，(3)代数圈和Hodge猜想，(4)混合Hodge模。(1)MT域是Hodge结构的分类空间，粗略地说，Mumford-Tate域的边界分量参数化了Hodge结构的退化。PI旨在通过研究Mumford-Tate域及其边界分量来促进数论、表示论和代数几何的发展。例如，PI计划扩展Carayol的工作，Carayol寻求将Galois表示与其阿基米德分量是离散级数的退化极限的自同构表示联系起来。(2)几何对象的模空间作为离散群的商的实现。这种实现的一个例子是非超椭圆亏格3曲线的模空间，它可以被实现为球商，其中所讨论的6维球位于K3曲面的MT域中。然而，已知的这种类型的例子并不多。私家侦探打算寻找更多。(3)第三个主题涉及利用Green和Griffiths的正规函数及其奇异性来研究Hodge猜想。私人投资促进局将在几个方向上发展这一方法。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将发展一种灵活的混合Hodge模的复变分理论，并将其应用于表示论中出现的问题。特别是，他们想要了解在稳定曲线的模空间上被视为复混合Hodge模的共形块的结构。Hodge理论是代数几何的中心领域，根源于经典的(19世纪)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簇的拓扑不变量与算术和解析不变量联系起来。中心概念是代数簇的上同调群上的Hodge结构。虽然上同调群是纯粹的拓扑性的，仅取决于簇的形状，但Hodge结构是一个更敏感的不变量。因此，霍奇结构承载了大量重要的代数几何和数论信息。代数几何中最著名的悬而未决的问题是Hodge猜想，这是一个关于簇的上同调群的Hodge结构与某些子簇的存在之间的关系的问题。这种对拓扑对象和更精细的解析不变量之间关系的关注是整个Hodge理论的典型，也是本FRG支持的研究的主要动机。因此，这项研究将对数论、代数几何和表示论等数学领域产生影响。由于涉及的技术数量较多，私人投资经理具有不同的技能和观点。FRG的一个重要组成部分将专门用于会议，这些会议将在私人投资机构之间交流思想，并就与霍奇理论有关的广泛主题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.