Mathematical Sciences: Variations on Hodge Structure

数学科学：Hodge 结构的变体

• 批准号: 9103203
• 负责人: Donu Arapura
• 金额: $ 4.76万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-01 至 1994-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9103203&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Variations; Hodge; Structure

This research will develop a relative Hodge theory of families of complex algebraic varieties. More precisely, there is associated to any variety a category of objects called stratified variations of Hodge structure. Given an algebraic map, the data of the cohomology along the fibers together with their mixed Hodge structures should give rise to such an object. The principal investigator will examine this situation, as well as consider the category of polarizable mixed Hodge structures. This is research in the field of algebraic geometry, which is one of the oldest parts of modern mathematics. In its origin, 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 also from analysis and topology. It is finding application in those areas of mathematics as well as in theoretical computer science and in robotics.

这项研究将发展一个复杂代数品种家族的相对杂物理论。 更确切地说，与任何变体相关联的一类对象类别称为霍奇结构的分层变化。 在给定代数图的情况下，沿纤维沿着纤维的共同体数据及其混合的船只结构的数据应引起这样的对象。 首席研究人员将检查这种情况，并考虑可极化混合霍奇结构的类别。 这是代数几何学领域的研究，这是现代数学最古老的部分之一。 它起源于它处理的数字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，该领域不仅利用了代数的方法，还利用了分析和拓扑。 它正在在数学以及理论计算机科学和机器人技术中找到应用。