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：
2016
资助国家：
加拿大
起止时间：
2016-01-01 至 2017-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615565
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.

我的研究将表示论、代数群、伽罗瓦上同调和代数几何等技术应用于域和曲线上代数群的 Torors 的各种问题。 1. tori的双有理不变理论：我们研究代数 k-tori（域上分割环面的扭转量）的（稳定）理性问题和更一般的（稳定）双有理分类问题以及有限群作用的分割环面商。自 70 年代 Voskresenskii 的开创性工作以来，是否所有稳定有理代数 k tori 都是有理的问题是一个有趣的问题。 Emmy Noether [N] 在她关于逆伽罗瓦问题的研究中首次研究了有限群上分裂圆环的商的合理性问题。我们还研究了配备有限群作用的分裂环面的（稳定）双有理等变线性化问题。这相当于询问有限群是否在第 n 个克雷莫纳群（射影 n 空间的双有理自同构群到线性自同构的子群）中共轭。克雷莫纳群是一个非常神秘的群：位面的克雷莫纳群是一个重要的研究对象；对于大于 3 的 n 知之甚少。 2. G 丛模堆的基本维数正如 Reichstein (2010 ICM)、Buhler 和 Merkurjev [BR,Re,Me] 所介绍的，基本维数是 G 轴的一个重要的数值不变量。通过 Dhillon，我们将有兴趣研究曲线上 G 丛模堆栈的基本维数，这是数学物理中的一个重要对象，概括了曲线上向量丛的模堆栈。 3. 霍尔代数和霍尔模：一个小的有限阿贝尔范畴的霍尔代数编码其对象的同构类之间的扩展结构。霍尔代数的重要例子是颤动的代数（林格尔，格林）[Ri，Gr]和曲线上的相干滑轮（卡普拉诺夫） [Kap2]（全部在有限域上考虑）。通过 Dhillon 和 Sala [PDF1]，我们有兴趣确定卡普拉诺夫霍尔代数投影线上辛/正交相干滑轮的霍尔模（由 Young 引入）。 4. 扭曲投影同质簇：扭曲投影同质簇是投影同质簇的形式。例子包括Severi Brauer簇（射影空间的扭曲形式）和广义Severi Brauer簇，Grassmannian的扭曲形式。扭曲射影齐次簇 Chow 群中的挠率问题是一个重要的问题，但尚未得到很好的理解。我们希望与 Junkins [PDF2] 和 Krashen 一起研究广义 Severi Brauer 品种的这个问题。早期的重要工作是由 Karpenko、Merkurjev [KM] 和 Karpenko [Kar] 在二次曲面上完成的； Baek [Ba] 关于 Severi Brauer 品种。