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，数论和表示论的问题。 研究人员打算集中在四个相关的主题：（1）Mumford-Tate（MT）域，（2）模空间，（3）代数圈和霍奇猜想，（4）混合霍奇模。 (1)MT域是Hodge结构的分类空间，而Mumford-Tate域的边界分量是Hodge结构退化的参数。 PI旨在通过研究Mumford-Tate域及其边界分量来推进数论，表示论和代数几何。 例如，PI计划扩展Carayol的工作，Carayol试图将Galois表示与自守表示相关联，自守表示的阿基米德分量是离散级数的退化极限。 (2)第二个主题是关于几何对象作为离散群的代数的模空间的实现。 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)第三个主题涉及的方法，霍奇猜想通过正常的功能和他们的奇异性，由于绿色和格里菲斯。PI将在几个方向上开发这种方法。 例如，他们将研究与正规函数相关联的阿基米德高度函数，并且他们打算研究与正规函数相关联的非约化MT群。 (4)最后，PI将开发一个灵活的混合霍奇模的复杂变化理论，并将其应用于表示论中出现的问题。 特别是，他们想了解的结构，共形块视为复杂的混合霍奇模的模空间的稳定curves.Hodge理论是一个中心领域的代数几何根源在古典（19世纪世纪）理论的特殊功能和周期积分。 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。 中心概念是霍奇结构的上同调群的代数品种。 虽然上同调群是纯拓扑的，只依赖于簇的形状，但霍奇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未解决问题是霍奇猜想，一个关于簇的上同调群的霍奇结构与某些子簇的存在之间关系的问题。 这种专注于拓扑对象和更精细的解析不变量之间的关系是典型的霍奇理论作为一个整体，它是主要动机的研究支持本联邦德国。 这项研究将因此影响数学的几个领域，包括数论，代数几何和表示论。 由于所涉及的技术的数量，PI有一套不同的技能和观点。 联邦德国的一个重要组成部分将致力于会议，这将交换PI之间的想法和培训博士后研究员和研究生在广泛的主题与霍奇理论。

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.