Mathematical Sciences: "Fundamental Groups Of Quasiprojective Varieties"

数学科学：“拟射影簇的基本群”

• 批准号: 9302531
• 负责人: Donu Arapura
• 金额: $ 6万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302531&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Fundamental; Groups; Quasiprojective

This award supports work on the fundamental group of a smooth quasiprojective variety defined over the field of complex numbers. If such a variety maps properly onto a hyperbolic curve, its fundamental group surjects onto a nonabelian free group. The principal investigator will determine to what extent the converse holds. The problem will be attacked by a variety of methods. These include a generalization of the Castenuovo-De Franchis lemma, the construction of L2 harmonic 1-forms on the universal cover, and a study of certain homologically defined subsets of the groups of 1-dimensional characters of the fundamental group. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.

该奖项支持在复杂数字领域定义的平滑准主体品种基本组的工作。 如果这样的品种正确地映射到双曲线曲线上，则其基本组将跃升至非阿比亚自由组。 首席研究人员将确定匡威保持在多大程度上。 该问题将受到多种方法的攻击。 这些包括对Castenuovo-de系列引理的概括，在通用覆盖层上的L2谐波1形构建以及对基本组的一维特征组的某些同源定义的子集的研究。 这是在代数几何学领域的重新搜索，这是现代数学的最古老部分之一，但在过去的四分之一世纪中，它具有革命性的开花。 在其起源中，它处理了可以通过最简单的方程（即多项式）在平面中定义的数字。 如今，该领域不仅利用了来自代数的方法，还利用了分析和拓扑的方法，相反，在这些领域以及理论计算机科学和机器人技术中找到了应用。