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

为了理解一个几何对象的数学性质，人们通常需要将性质相似的对象族组合在一起，并将一族作为一个整体来研究。这个概念在量子物理学和弦理论中也很自然地出现。这个研究项目的很大一部分是致力于理解的几何属性的参数空间间接继承的对象，它参数化。主要研究者计划研究这些家庭的普遍性质方面的各种概念的负曲率。有些问题要研究的是长期存在的代数和复杂的几何。另一个研究方向是对一些特殊的几何对象进行分类，这些对象被称为伪紧埃尔米特对称空间。这些对象提供了有趣的几何模型，用于研究来自不同数学领域的广泛问题，包括代数几何，微分几何，数论和表示论。更具体地说，首席研究员将在代数和复杂几何的几个方向进行研究。在该项目的第一部分，他打算应用最近开发的工具，包括广义Weil-Petersson度量来理解特定极化流形族的各种双曲性性质，包括Kahler Ricci平坦流形族的双曲性，特殊对数一般型流形族和一般型流形族。特别是，他将调查一个猜想Viehweg关于日志一般型性质的模空间的规范极化流形。在第二部分中，他计划完成一个关于算术伪紧埃尔米特对称空间分类的合作项目，这是早期关于伪投影平面的工作的自然延续。此外，他还将研究与这些假结构相关的问题，例如从派生类别的角度对假投影平面上的异常集合进行调查，对复杂奇异二次曲面或假二次曲面的普遍覆盖进行表征，以及对Cartwright-Steger曲面几何特性的理解。

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.