Algebraic and Geometric Aspects of Algebraic Groups and Homogeneous Varieties

代数群和齐次簇的代数和几何方面

基本信息

批准号：
RGPIN-2016-05215
负责人：
Lemire, Nicole
金额：
$ 1.31万
依托单位：
University of Western Ontario
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2019
资助国家：
加拿大
起止时间：
2019-01-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676833
关键词：
Algebraic Geometric Aspects Groups Homogeneous

项目摘要

My research applies techniques from representation theory, algebraic groups, Galois cohomology and algebraic geometry to a variety of problems about torsors for algebraic groups over fields and curves.******1. Birational Invariant theory for tori:***We study (stable) rationality problems and more generally (stable) birational classification problems for algebraic k-tori (torsors for split tori over a field) and quotients of split tori by finite group actions. The question of whether all stably rational algebraic k tori are rational is an interesting question, open since Voskresenskii's seminal work in the 70s. The rationality problem for quotients of split tori over a finite group was first studied by Emmy Noether [N] in her work on the inverse Galois problem.******We also study the (stable) birational equivariant linearisation problem for split tori equipped with the action of a finite group. This is equivalent to asking whether the finite group is conjugate in the nth Cremona group, the group of birational automorphisms of projective n space to a subgroup of linear automorphisms. The Cremona group is a very mysterious group: the Cremona group of the plane is an important object of study; not much is known for n bigger than 3.******2. Essential Dimension of Moduli stacks of G bundles***Essential dimension is an important numerical invariant of G torsors, as introduced by Reichstein (2010 ICM), Buhler and Merkurjev [BR,Re,Me]. With Dhillon, we will be interested in studying the essential dimension of the moduli stack of G-bundles over a curve, an important object in mathematical physics, generalising the moduli stack of vector bundles over a curve. ******3. Hall Algebras and Hall modules:***A Hall algebra of a small finitary abelian category encodes the structure of extensions between isomorphism classes of its objects.Important examples of Hall algebras are those of quivers (Ringel, Green) [Ri,Gr] and for coherent sheaves over a curve (Kapranov) [Kap2] (all considered over a finite field). With Dhillon and Sala [PDF1], we are interested in determining the Hall module (introduced by Young) of symplectic/orthogonal coherent sheaves over the projective line over Kapranov's Hall algebra.***4. Twisted projective homogeneous varieties: Twisted projective homogeneous varieties are forms of projective homogeneous varieties. Examples include Severi Brauer varieties (twisted forms of projective space) and Generalised Severi Brauer varieties, twisted forms of Grassmannians. The question of torsion in Chow groups of twisted projective homogeneous varieties is an important one and not well understood. With Junkins [PDF2] and Krashen, we wish to study this question for generalised Severi Brauer varieties. Important earlier work was done by Karpenko, Merkurjev [KM] on quadrics, and Karpenko [Kar]; Baek [Ba] on Severi Brauer varieties.*** **

我的研究应用了代表理论，代数群体，Galois的共同体和代数几何形状的技术，到有关代数群体和曲线的Torsors的各种问题。****** 1。托里的异常不变理论：***我们研究（稳定的）合理性问题，更普遍地（稳定）代数k-tori的（稳定）的（稳定的）偶然分类问题（在田野上分裂的托里（Tori）的托架）和有限的小组动作的分裂托里（Tori）的商。自从Voskresenskii在70年代的开创性工作以来，所有稳定的理性代数K Tori是否都是理性的问题。艾米（Emmy）noether [n]在她在逆galois问题上的工作中首先研究了有限群体的分裂托里商的合理性问题。配备有限组的行动。这相当于询问有限的组是否在nth Cremona组中是共轭的，这是射影n空间的异态自动形态的组与线性自动形态的亚组。 Cremona组是一个非常神秘的团体：飞机的Cremona群体是研究的重要对象； n大于3。****** 2。 G束的模量堆栈的基本维度***基本维度是G Torsors的重要数值不变性，如Reichstein（2010 ICM），Buhler和Merkurjev [BR，RE，RE，ME]所引入的。借助Dhillon，我们将有兴趣研究曲线上G型束模量的基本维度，曲线是数学物理学中的重要对象，从而概括了曲线上的向量捆绑包的模量。 ****** 3。霍尔代数和霍尔模块：***一个小粉饰阿贝尔类别的大厅代数编码其对象的同构类别之间的扩展结构。大厅代数的示例是Quivers（Ringel，Green）[RI，GR]和对于曲线（Kapranov）[Kap2]（全部考虑在有限场）上的相干滑轮。借助Dhillon和Sala [PDF1]，我们有兴趣确定Kapranov的Hall Algebra上的投影线上的Simphectic/正交相干支线（由Young介绍）（由Young引入）。*** 4。扭曲的投射均匀品种：扭曲的投射均匀品种是投射均匀品种的形式。例子包括Severi Brauer品种（投影空间的扭曲形式）和广义的Severi Brauer品种，以及扭曲的司羊毛人。扭曲的投影型均匀品种中的扭转问题是一个重要的扭曲问题，并且不太了解。与Junkins [PDF2]和Krashen一起，我们希望研究这个问题，以供广义的Severi Brauer品种。重要的早期工作是由Karpenko，Merkurjev [KM]和Karpenko [Kar]完成的； baek [ba] on Severi Brauer品种。*** **