Hodge theory of toroidal compactifications and Torelli theorems — K3 surfaces,abelian varieties, and IHSM

在这个项目中，我们想要研究模空间的紧化和Hodge理论之间的相互作用。我们主要关注交换簇、K3曲面和不可约全纯辛流形(IHSM)的模空间。代数簇的分类是代数几何中最基本的任务之一。这通常导致模空间的构造，该模空间将待分类的对象参数化。K3曲面和阿贝尔簇是研究最深入的代数簇之一，在过去的几十年里，作为K3曲面的高维模拟的IHSM也成为代数几何中的一个反复出现的话题。这类簇的模空间总是拟投射的，但通常不是投射的，这是模理论中的一个中心问题，要求这种模空间具有良好的紧性。Hodge理论与代数几何之间的关系是经典的，它建立在代数簇的周期的计算上。尽管霍奇理论是一种基本的分析理论，但它的主要应用是代数和算术几何。Torelli定理建立了模空间的几何与周期映射的像之间的联系，在我们感兴趣的情况下，周期映射甚至是满足性的。因此，人们可以通过算术群来识别具有周期域的商的这种模空间。这赋予了这些空间丰富的结构，并允许我们使用李理论和模形式的技巧。特别是，由于芒福德、洛伊金加和其他理论，我们有环形或半立方压实理论可供我们使用。这一点已在本项目的第一部分中得到了利用，并导致了K3曲面的模空间的新的半群紧化。在本项目中，我们将使用Torelli定理作为指导原则。更准确地说，我们希望通过Torelli定理将簇的退化(应该对应于模空间的紧化中的边界点)与Hodge结构的退化(应该联系到特定的半格紧化)联系起来。本研究项目的最终目的是利用Hodge理论构造模空间的模紧化。

In this project we want to investigate the interplay between compactifications of moduli spaces and Hodge theory. Our main focus are moduli spaces of abelian varieties, K3 surfaces, and irreducible holomorphic symplectic manifolds (IHSM). The classification of algebraic varieties is one of the most fundamental tasks in algebraic geometry. This typically leads to the construction of moduli spaces which parametrize the objects to be classified. K3 surfaces and abelian varieties are among the most thoroughly studied classes of algebraic varieties and in the last decades also IHSM, which are higher dimensional analogs of K3 surfaces, have become a recurrent topic in algebraic geometry. The moduli spaces of these classes of varieties are always quasi-projective, but usually not projective and it is a central question in moduli theory to ask for good compactifications of such moduli spaces.The relation between Hodge theory and algebraic geometry is classical and was built around the computation of periods of algebraic varieties. Even though Hodge theory is a fundamentally analytic theory, its main applications are to algebraic and arithmetic geometry. Torelli theorems establish a link between the geometry of moduli spaces and the images of period maps and in the cases of interest to us the period maps are even surjective. As a result one can identify such a moduli space with a quotient of the period domain by an arithmetic group. This endows these spaces with a rich structure and allows us to use techniques from Lie theory and modular forms. In particular, we have the theory of toroidal or semitoric compactifications due to Mumford, Looijenga, and others at our disposal. This has been exploited in the first part of this project and has led, among other things, to new semitoric compactifications of moduli spaces of K3 surfaces.In the current project we will use Torelli theorems as the guiding principle. More precisely, we want to relate degenerations of varieties (which should correspond to boundary points in a compactification of the moduli spaces) to degenerations of Hodge structures (which should be linked to specific semitoric compactifications) via Torelli theorems. The ultimate goal of this research project is the construction of modular compactifications of moduli spaces by means of Hodge theory.