Brill-Noether Theory for K3 Surfaces; Compactified Picards and Coadjoint Orbits of Loop Algebras

K3 曲面的布里尔-诺特理论；

• 批准号: 9802532
• 负责人: Eyal Markman
• 金额: $ 7.06万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802532&HistoricalAwards=false
• 关键词: Brill; Noether; Theory; K3; Surfaces

Markman 9802532 The problems studied in this project arise from the interaction of algebraic geometry, symplectic geometry and integrable systems. Part 1 of the project consists of the study of a birational duality among moduli spaces of sheaves on K3 surfaces arising from the Brill-Noether stratification of these symplectic moduli spaces. The interaction of this duality with a representation of the Heisenberg algebra on the total cohomology of sequences of moduli spaces is investigated. In Part 2 of the project the theory of integrable systems will be used to compare the boundary of the compactified Picard of an algebraic curve with planar singularities to the boundary of coadjoint orbits of loop algebras. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. 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 from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

本课题研究的问题源于代数几何、辛几何和可积系统的相互作用。项目的第一部分包括研究由这些辛模空间的Brill-Noether分层引起的K3表面上的轴模空间之间的双国对偶性。研究了这种对偶性与模空间序列全上同调上的海森堡代数表示的相互作用。在项目的第二部分中，将使用可积系统理论来比较具有平面奇点的代数曲线的紧化皮卡德边界与环代数的伴随轨道边界。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。