Generic Flag Varieties

通用旗帜品种

• 批准号: RGPIN-2020-04008
• 负责人: Karpenko, Nikita
• 金额: $ 1.53万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758346
• 关键词: Generic; Flag; Varieties

Generic Flag Varieties It is a very common situation in Algebra that a type of objects is considered over field extensions of some fixed base field. Among the objects of the given type, there is usually a particularly simple one -- the split object. Its properties are well-understood and the problems of interest are easily solved for it. The nature of a general object of the given type is usually too complicated to be addressed directly. An important step towards its understanding is the study of the so-called generic object. This can be considered as the farthest opposite to the split one. It is characterized by the property that every object is its specialization. The use of generic objects is an important ingredient in many proofs. Surprisingly, generic objects are more accessible than the general (i.e., arbitrary) ones. At the same time, information on generic objects provides us with some important information on the general objects. Also generic objects provide a testing ground for general conjectures. This project suggests the study of generic objects arising from algebraic groups as described below. It is again a common situation in Algebra that objects of the given type are classified by G-torsors (or principle G-homogeneous spaces), for certain split semisimple algebraic group G. In such situations, the study of generic objects becomes the study of generic G-torsors. The main objective of the present project is to achieve a complete understanding of the Chow ring CH(E/B), where B is a Borel subgroup of G. The quotient variety E/B is a smooth projective variety known as a generic flag variety. The Grothendieck ring K(E/B) of the variety E/B is endowed with the filtration by codimension of support. Let GK(E/B) be the associated graded ring. Main Question on the Chow ring CH(E/B) asks whether the canonical surjective homomorphism of graded rings CH(E/B) onto GK(E/B) is an isomorphism. Equivalently, it asks if the connective K-theory of E/B is free of torsion. In this project, I want to study this question and some related problems. A breakthrough in this direction will be a major achievement in the field of algebraic groups. The question has already been answered by positive for several types of G. As a consequence, the Chow ring as well as the connective K-theory are completely understood for such G. On the other hand, it has been recently shown that the answer is negative for the spinor group Spin(17). One of the related problems is the problem of comparison of the Chow ring of the classifying space BG of G with the representation ring R(G) viewed as the K-theory of BG. In a recent joint work with A. Merkurjev, this problem has been approached by introducing an analogue of the filtration by codimension of support on R(G). For further progress, we plan to construct a G-equivariant version of the connective K-theory and, in particular, the connective K-theory of BG.

通用标志种类 在代数中一种非常常见的情况是，在某些固定基域的域扩展上考虑一种对象类型。在给定类型的对象中，通常有一种特别简单的对象——分割对象。它的属性很容易理解，并且很容易解决感兴趣的问题。给定类型的一般对象的性质通常太复杂而无法直接解决。理解它的一个重要步骤是对所谓的通用对象的研究。这可以认为是与分裂最远的相反。它的特点是每个对象都是它的特化。通用对象的使用是许多证明中的重要组成部分。令人惊讶的是，通用对象比一般（即任意）对象更容易访问。同时，通用对象的信息为我们提供了一些关于通用对象的重要信息。通用对象还为一般猜想提供了试验场。该项目建议研究由代数群产生的通用对象，如下所述。对于某些分裂半简单代数群 G，给定类型的对象通过 G-torsors（或原理 G-齐次空间）进行分类，这也是代数中的常见情况。在这种情况下，对通用对象的研究就变成了对通用 G-torsor 的研究。本项目的主要目标是实现对 Chow 环 CH(E/B) 的完整理解，其中 B 是 G 的 Borel 子群。商簇 E/B 是平滑射影簇，称为通用旗簇。 E/B品种的格洛腾迪克环K(E/B)通过支撑余维被赋予过滤作用。令 GK(E/B) 为相关的分级环。关于 Chow 环 CH(E/B) 的主要问题询问分级环 CH(E/B) 到 GK(E/B) 上的规范满射同态是否是同构。等效地，它询问 E/B 的连接 K 理论是否没有挠率。在这个项目中，我想研究这个问题以及一些相关的问题。这个方向的突破将是代数群领域的重大成就。对于几种类型的 G，这个问题已经得到了肯定的答案。因此，对于这种 G，Chow 环以及联结 K 理论是完全可以理解的。另一方面，最近表明，对于旋量群 Spin(17)，答案是否定的。相关问题之一是 G 的分类空间 BG 的 Chow 环与被视为 BG 的 K 理论的表示环 R(G) 的比较问题。在最近与 A. Merkurjev 的联合工作中，通过引入 R(G) 上支持余维的过滤模拟来解决这个问题。为了进一步取得进展，我们计划构建联结 K 理论的 G 等变版本，特别是 BG 的联结 K 理论。