周期映射与模空间作为球商的刻画

本项目研究模空间作为球商的刻画，主要工具是形变理论与Hodge结构的变形。模空间是一类射影流形组成的空间，具有特定的几何结构。周期域是一类Hodge分解组成的集合，具有光滑的拟射影代数簇的结构。周期映射是从模空间出发到周期域的全纯映射，它将模空间中的每个射影流形映射到该流形上的Hodge分解。Hodge结构的变形等价于一个周期映射。本项目将通过研究射影流形上Hodge分解的性质，来证明在特定条件下周期域中有一个同构于复欧氏空间中单位开球的子集，使得周期映射的像落在该子集中。这样，在整理Torelli定理的前提下（即周期映射是单射），我们得到模空间是球商的子集，这里的球商即开球在单值群作用下的商空间。本项目的关键点就是Calabi-Yau型流形中Hodge丛的典则横截的渐进展开，这是近年来Calabi-Yau流形相关结果的推广。

This project is aimed at characterization of moduli spaces as ball quotients, for which the main tools are deformation theory and variation of Hodge structure. The moduli space is the space of projective manifolds of certain type, which is endowed with some geometric structure. The period domain is the set of all the Hodge decompositions of certain type, which is a smooth quasi-projective variety. Then the period map is defined as a holomorphic map from the moduli space to the period domain, which maps each projective manifold in the moduli space to its Hodge decomposition. The variation of Hodge structure is equivalent to the period map. By studying the properties of the Hodge decomposition of each projective manifold in the moduli space, we will show under certain conditions that there exists a subset of the period domain, which is isomorphic to the unit open ball in the complex Euclidean space, such that the image of the period map lies in this subset. Then, under the global Torelli theorem, which is equivalent to that the period map is injective, we conclude that the moduli space is a subset of the open ball quotient by the monodromy group, which we call ball quotient in this project. The key point of this project is the expansion of the canonical section of the Hodge bundle for Calabi-Yau type manifolds, which is a generalization of that for Calabi-Yau manifolds in recent years.