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：二分几何和微分几何。具体地说，它处理在代数几何和微分几何中存在多少具有给定性质的几何对象的自然问题。这样的问题被称为“代数变异类的有界性”。该项目分为两个部分，其中PI将特别致力于理解与某些种类的有界性相关的不同几何方面（定义为多项式方程零轨迹的几何对象）以及这些结果的应用。其主要目的是开发一种方法，可以将低维几何的经典结果扩展到高维几何，使用过去20年发展起来的被称为“最小模型程序”的理论技术。其中一些结果将应用于数学物理中的弦理论。项目的第一部分侧重于卡拉比-丘品种的有界性。尽管Calabi-Yau流形从许多角度看都是非常简单的代数变体，但人们对其有界性的认识仍然很少。该项目的主要目的是研究椭圆纤维的葫芦品种。已知在低维它们满足有趣的有界性，PI打算在任意维上推广这一结果。这些结果在最小模型规划、代数变异的分类和f理论中都有应用。本项目的最后一部分旨在探索具有锥边奇点的卡勒-爱因斯坦度量理论的类似技术的含义。这个项目的主要动机是研究球商的环面紧化和更一般的高阶局部对称变体的双民族几何。作为该项目的一部分，PI打算开发一种技术，该技术最终将适用于具有压缩负截面曲率的有限体积Kahler流形的紧化。

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.