Geometry of analytic and algebraic varieties

这个研究项目是在代数几何领域，这是纯数学的核心领域之一，其根源可以追溯到古希腊。代数几何的核心是研究多项式方程解的几何形式。这些是最简单的方程，因此，毫不奇怪，它们在几乎任何科学学科中都发挥着重要作用。特别是代数几何与微分和解析几何、交换代数、拓扑学、数论、物理学、理论计算机科学、密码学和许多应用数学领域有着密切的联系。代数几何中最基本的问题是对所有由多项式方程定义的几何对象进行分类。最小模型程序是这种分类程序中最成功的方法，最近在分类复杂品种（即由复数组成的解集）方面取得了非凡的成功。希望这些技术将扩展到其他环境，如在正特征域上定义的品种和复杂的分析品种。研究生和博士后将参与该项目的各个方面。这个项目将把复数上的最小模型程序的结果推广到正特征的代数闭域上的品种（以及混合特征的情况）以及Kahler品种的情况。特别是，PI将表明最小模型程序适用于紧Kahler klt的三倍和四倍，并将发展广义klt Kahler变分的理论。PI将研究特征3的三倍的最小模型程序和大特征代数闭域上射影三倍的多正则映射。本文还将开发混合特性的最小模型程序。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project is in the field of algebraic geometry, one of the core areas of pure mathematics whose roots date back to the ancient Greeks. At its heart, algebraic geometry studies the geometry of the solution of polynomial equations. These are the simplest possible equations and so, not surprisingly, they play an important role in almost any scientific discipline. In particular algebraic geometry has close ties with differential and analytic geometry, commutative algebra, topology, number theory, physics, theoretical computer science, cryptography, and many areas of applied mathematics. The most fundamental problem in algebraic geometry is to classify all geometric objects defined by polynomial equations. The minimal model program is the most successful approach to this classification program and has recently had extraordinary success in classifying complex varieties, i.e. solution sets consisting of complex numbers. It is hoped that these techniques will extend to other contexts such as varieties defined over fields of positive characteristics and to complex analytic varieties. The PI will involve graduate students and post-docs in various aspects of this project.This project will generalize the results of the minimal model program over the complex numbers to the case of varieties over algebraically closed fields of positive characteristic (and to the case of mixed characteristics) as well as the case of Kahler varieties. In particular, the PI will show that the minimal model program holds for compact Kahler klt threefolds and fourfolds and will develop the theory of generalized klt Kahler varieties. The PI will investigate the minimal model program for threefolds in characteristic 3 and the pluricanonical maps of projective 3-folds over algebraically closed fields of large characteristics. The minimal model program in mixed characteristic will also be developed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.