Hyperbolic Properties of Families of Polarized Manifolds and Problems Related to Fake Compact Hermitian Symmetric Spaces

为了理解一个几何物体的数学性质，人们通常需要将性质相似的物体组合在一起，并将其作为一个整体进行研究。这个概念在量子物理学和弦理论中也很自然地出现。该研究项目的很大一部分致力于理解间接继承自其参数化对象的参数空间的几何性质。首席研究员计划根据曲率的各种负性概念来研究这些族的普遍性质。要研究的一些问题是代数和复杂几何中长期存在的问题。另一个研究方向是对一些特殊几何对象的分类，即假紧致厄米对称空间。这些对象为研究来自不同数学领域的广泛问题提供了有趣的几何模型，包括代数几何、微分几何、数论和表示理论。更具体地说，首席研究员将在代数和复杂几何的几个方向上进行研究。在项目的第一部分，他打算应用最近开发的工具，包括广义的Weil-Petersson度量来理解极化流形的特定族的各种双曲性，包括Kahler Ricci平面流形族的双曲性，特殊对数一般型流形族的双曲性，以及一般型流形族的双曲性。特别地，他将研究Viehweg关于正则极化流形模空间的对数一般型性质的一个猜想。在第二部分，他计划完成一个关于算法伪紧厄米对称空间的分类的合作项目，这是早期关于伪投影平面的工作的自然延续。此外，他将研究与这些假结构相关的问题，例如从衍生范畴的角度研究假投影平面上的特殊集合，复杂奇异二次曲面或假二次曲面的普遍覆盖的表征，以及对Cartwright-Steger曲面几何性质的理解。

To understand the mathematical properties of a geometric object, often one needs to group families of objects of similar nature together and study a family as a whole. This concept arises naturally in quantum physics and string theory as well. A large part of this research project is devoted to the understanding the geometric properties of the parameter space inherited indirectly from the objects that it parameterizes. The Principal Investigator plans to investigate universal properties of such families in terms of various notions of negativity in curvature. Some of the problems to be studied are longstanding ones in algebraic and complex geometry. Another line of research is on the classification of some special geometric objects known as fake compact Hermitian symmetric spaces. These objects provide interesting geometric models for studying a wide range of problems from different areas of mathematics, including algebraic geometry, differential geometry, number theory, and representation theory. More specifically, the Principal Investigator will pursue research in several directions in algebraic and complex geometry. In the first part of the project, he intends to apply recently-developed tools involving generalized Weil-Petersson metrics to understand various hyperbolicity properties of specific families of polarized manifolds, including the hyperbolicity for families of Kahler Ricci flat manifolds, families of special log-general type manifolds, and families of general-type manifolds. In particular, he will investigate a conjecture of Viehweg about log-general type properties of moduli spaces of canonically polarized manifolds. In the second part, he plans to complete a collaborative project on the classification of arithmetic fake compact Hermitian symmetric spaces, a natural continuation of earlier work on fake projective planes. In addition, he will study problems related to these fake structures, such as the investigation of exceptional collections on fake projective planes from the point of view of derived categories, the characterization of the universal covering of a complex exotic quadric or a fake quadric, and the understanding of geometric properties of Cartwright-Steger surfaces.