Effective Boundedness Results in Algebraic Geometry

这个项目的目的是研究几何的不同方面之间的联系：双有理几何和微分几何。具体来说，它涉及自然问题有多少几何对象拥有给定的属性所产生的代数几何和微分几何确实存在。这类问题被称为“代数簇类的有界性”。“该项目分为两个部分，其中PI将特别旨在了解与某些类别的品种（定义为多项式方程零轨迹的几何对象）的有界性相关的不同几何方面以及这些结果的应用。其主要目的是发展的方法，可用于扩展的经典结果，在低维几何更高的维度，使用技术所产生的发展，在过去的20年中被称为“最小模型计划”的理论。“其中一些结果将应用于数学物理学中的弦理论。该项目的第一部分集中在卡-丘簇的有界性。尽管卡-丘流形在许多观点上都是非常简单的代数簇，但它们的有界性仍然知之甚少。这部分的主要目标是研究椭圆纤维的Calabi-Yau簇。它是已知的，在低维，他们满足有趣的有界性的性质和PI打算推广这样的结果在任意维。这些结果在极小模型规划、代数簇的分类和F-理论中有应用。本计画的最后一部分是探讨类似的技巧对锥边奇异的Kahler-Einstein度规理论的意涵。这个项目的主要动机是研究球对称的环面紧化的双有理几何和更一般的高阶局部对称簇。作为这个项目的一部分，PI打算开发的技术，最终将工作，甚至与紧致有限体积Kahler流形捏负截面曲率。

The purpose of this project is to study connections between different aspects of geometry: birational geometry and differential geometry. Specifically, it deals with natural questions on how many geometric objects possessing given properties arising both in algebraic geometry and differential geometry do exist. Such questions are known as "boundedness of classes of algebraic varieties." The project is divided in two parts in which the PI will especially aim to understand different geometrical aspects related to the boundedness of certain classes of varieties (geometric objects defined as zero loci of polynomial equations) and applications of such results. The main aim is to develop methods that can be used to extend classical results in low-dimensional geometry to higher dimensions using techniques arising from the developed in the last 20 years theory known as the "minimal model program." Some of these results will have application to string theory in mathematical physics.The first part of the project focuses on boundedness of Calabi-Yau varieties. Even though Calabi-Yau manifolds are very simple algebraic varieties from many points of view, their boundedness properties are still poorly understood. The main goal of this part of the project is to study elliptic fibered Calabi-Yau varieties. It is known that in low dimensions they satisfy interesting boundedness properties and the PI intends to extend such results in arbitrary dimension. These results have applications to the minimal model program, the classification of algebraic varieties and F-theory. The last part of this project aims to explore the implications of similar techniques to the theory of Kahler-Einstein metric with cone-edge singularities. The main motivation for this project is the study of the birational geometry of toroidal compactifications of ball quotients and more generally of higher ranks locally symmetric varieties. As part of this project, the PI intends to develop techniques that eventually will work even with compactifications of finite volume Kahler manifolds with pinched negative sectional curvature.