Problems in Equivariant Algebraic Geometry

等变代数几何问题

• 批准号: 0101543
• 负责人: William Graham
• 金额: $ 6.41万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-10-15 至 2004-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101543&HistoricalAwards=false
• 关键词: Problems; Equivariant; Algebraic; Geometry

The purpose of this project is to use equivariant methods to studyproblems concerning algebraic varieties with group actions. Partof this project involves moment graphs of varieties with torusactions. For a variety on which a torus acts with finitely manyfixed points and curves, one can define a combinatorial objectcalled a moment graph. There has been much recent progress incomputing topological invariants, such as cohomology orintersection homology, in terms of the moment graph. Theinvestigator proposes to extend this to K-theory. Schubertvarieties are among the most important varieties with this type oftorus action; the investigator plans to continue work with BrianBoe on a conjecture that would simplify determining if a point ina Schubert variety is rationally smooth. Moreover, he plans toextend some facts known only for Schubert varieties to the moregeneral setting of varieties with this type of torus action. Inaddition, the investigator plans to use equivariant methods tostudy certain interesting varieties: he plans to calculateChern-Schwartz-MacPherson classes of degeneracy loci, and tocalculate interesting invariants (degrees, push-forward measures)of nilpotent adjoint orbits of reductive Lie groups.This project is in the area of mathematics referred to as"algebraic geometry." Algebraic geometry studies geometricobjects by describing them as solutions to polynomial equations-- such objects are called "algebraic varieties." Fortunately,although algebraic varieties can be very complicated, many ofthem have a great deal of symmetry. Mathematicians have beenintensely investigating such varieties with extra symmetry forseveral reasons: The presence of such extra symmetry makesthese varieties easier to study, so that these varieties arevaluable test cases in developing methods to investigate allvarieties. Moreover, varieties with extra symmetry are of greatinterest in their own right: they play important roles invarious areas of mathematics, including number theory,combinatorics, and representation theory. There has beenconsiderable recent progress in developing techniques tounderstand these kinds of varieties; this project is aboutextending these techniques, and using them to study particularclasses of varieties.

本课题的目的是利用等变方法研究具有群作用的代数簇的问题。 该项目的一部分涉及具有环面作用的品种的矩图。 对于一个簇，其上的环面有许多不动点和曲线，人们可以定义一个组合对象，称为矩图。 利用矩图计算拓扑不变量，如上同调、交同调等，近年来取得了很大的进展。 研究者建议将其扩展到K理论。 舒伯特品种是其中最重要的品种与这种类型oftorus行动;调查计划继续工作与布赖恩波的猜想，将简化确定一个点在舒伯特品种是合理的顺利。 此外，他还计划将一些只为舒伯特簇所知的事实推广到具有这种环面作用的更一般的簇的情况。 此外，研究人员计划使用等变方法来研究某些有趣的变种：他计划计算退化轨迹的Chern-Schwartz-MacPherson类，并计算约化李群的幂零伴随轨道的有趣不变量（度，推进测度）。代数几何通过将几何对象描述为多项式方程的解来研究它们--这种对象被称为“代数簇“。幸运的是，虽然代数簇可能非常复杂，但它们中的许多都具有很强的对称性。 数学家们一直在研究具有额外对称性的变种，原因有几个：这种额外对称性的存在使这些变种更容易研究，因此这些变种在开发研究所有变种的方法时是有价值的测试案例。此外，具有超对称性的变体本身就具有很大的意义：它们在数学的各个领域，包括数论、组合学和表示论，都扮演着重要的角色。最近在发展了解这些品种的技术方面取得了相当大的进展;本项目是关于扩展这些技术，并利用它们来研究特定类别的品种。