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

要了解几何对象的数学性质，通常需要将性质相似的对象的族组合在一起，并将族作为一个整体进行研究。这个概念自然而然地出现在量子物理学和弦理论中。这个研究项目的很大一部分致力于理解参数空间的几何性质，这些几何性质是从参数空间的对象间接继承而来的。首席调查员计划根据曲率中负性的各种概念来研究这类家庭的普遍性质。其中一些要研究的问题是代数和复杂几何中的长期问题。另一项研究是关于某些特殊几何对象的分类，这些几何对象被称为伪紧Hermitian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各种问题，包括代数几何、微分几何、数论和表示论。更具体地说，首席调查员将在代数和复杂几何方面进行多个方向的研究。在项目的第一部分，他打算利用最近发展起来的涉及广义Weil-Petersson度量的工具来了解特定极化流形家族的各种双曲性性质，包括Kahler Ricci平坦流形家族、特殊对数通用型流形家族和一般类型流形家族的双曲性。特别是，他将研究Viehweg关于标准极化流形的模空间的对数通用型性质的猜想。在第二部分中，他计划完成一个关于算术伪紧Hermitian对称空间分类的合作项目，这是早期关于伪射影平面工作的自然延续。此外，他还将研究与这些伪结构有关的问题，如从派生范畴的角度考察伪射影平面上的例外集合，刻画复奇异二次曲面或伪二次曲面的泛覆盖，以及理解Cartwright-Steger曲面的几何性质。

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.