Geometry of Deligne-Lusztig varieties

Deligne-Lusztig 品种的几何形状

• 批准号: 171349465
• 负责人: Professor Dr. Ulrich Görtz
• 金额: --
• 依托单位: Institut für Experimentelle Mathematik (IEM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171349465?language=en
• 关键词: Geometry; Deligne; Lusztig; varieties

Representation theory, in other words the study of symmetries, is an important area of mathematics. For a certain class of “representations” Pierre Deligne and George Lusztig exhibited a close connection between the algebraic notion of representation, and certain geometric objects, now called Deligne-Lusztig varieties. Since then, Deligne-Lusztig varieties have been thoroughly investigated from a representation theoretic point of view. Much less is known about their geometric properties, though. The goal of the proposed project is to improve our understanding of the geometry of Deligne-Lusztig varieties using explicit and algorithmic methods in order to study concrete examples, and hopefully eventually to reach new general results. Deligne-Lusztig varieties are smooth quasi-projective varieties over finite fields. An important specific question is whether there exist Deligne-Lusztig varieties which are not affine varieties.

表示论，换句话说，研究对称性，是数学的一个重要领域。对于某一类“表象”，皮埃尔·德利涅和乔治·卢斯蒂格展示了表象的代数概念与某些几何对象（现在称为德利涅-卢斯蒂格簇）之间的密切联系。从那时起，Deligne-Lusztig簇从表示论的角度进行了深入的研究。然而，人们对它们的几何性质知之甚少。该项目的目标是提高我们的理解几何的Deligne-Lusztig品种使用明确的和算法的方法，以研究具体的例子，并希望最终达到新的一般结果。Deligne-Lusztig簇是有限域上的光滑拟投射簇。一个重要的具体问题是，是否存在Deligne-Lusztig变种，而不是仿射变种。