Effective Boundedness Results in Algebraic Geometry

代数几何中的有效有界性结果

• 批准号: 1817309
• 负责人: Gabriele Di Cerbo
• 金额: $ 11万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-16 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1817309&HistoricalAwards=false
• 关键词: Effective; Boundedness; Results; Algebraic; Geometry

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.

该项目的目的是研究几何不同方面之间的联系：双有理几何和微分几何。具体来说，它解决了代数几何和微分几何中存在多少具有给定属性的几何对象的自然问题。此类问题被称为“代数簇类的有界性”。该项目分为两个部分，其中 PI 将特别致力于理解与某些类簇（定义为多项式方程零轨迹的几何对象）有界性相关的不同几何方面以及这些结果的应用。主要目标是开发可用于将低维几何中的经典结果扩展到更高维度的方法，利用过去 20 年开发的理论（称为“最小模型程序”）中产生的技术。其中一些结果将应用于数学物理中的弦理论。该项目的第一部分重点关注 Calabi-Yau 簇的有界性。尽管卡拉比-丘流形从许多角度来看都是非常简单的代数簇，但它们的有界性质仍然知之甚少。这部分项目的主要目标是研究椭圆纤维 Calabi-Yau 品种。众所周知，在低维度中它们满足有趣的有界属性，并且 PI 打算将这样的结果扩展到任意维度。这些结果可应用于最小模型程序、代数簇分类和 F 理论。该项目的最后一部分旨在探索类似技术对具有锥边奇点的卡勒-爱因斯坦度量理论的影响。该项目的主要动机是研究球商环形紧化的双有理几何，以及更普遍的高阶局部对称簇的双有理几何。作为该项目的一部分，PI 打算开发技术，最终即使在压缩负截面曲率的有限体积卡勒流形的情况下也能发挥作用。