Incompressibility of algebraic varieties

该方案的主要目的是研究扭量的数值不变量和离散不变量以及 射影齐次簇，如本质维度和标准维度及其 在非闭域上代数群理论及相关结构中的应用。 利用兼容的本地数据和自然环境构建全局信息/对象的想法 这样做的障碍，对数学和物理(流形)是非常重要的 是由局部拼凑而成的全局对象的典型例子 数据)。流形的代数版本(例如在数论的应用中需要， 但也在遥远的领域，如遗传学)被称为品种，或更一般的方案。 扭矩理论提供了测量修补障碍的语言和工具 将本地数据放在一起(例如构建各种数据或方案)。本质维度理论 可以被认为是扭矩的一种“复杂性理论”。与此相关的概念是 规范维度和可压缩变化(一种工具，允许我们知道何时确定 构造的类型再简单不过了)。在介绍了我的核心部分后 现在，我将对我目前的一些研究进行更详细的描述 和目标。 一个代数簇称为不可压缩的，如果它的所有有理自同态都是占优的。 关于变元不可压缩性的结果有着广泛的应用，特别是在计算方面 代数结构的本质维度。几类簇的不可压缩性 已由申请人设立。大多数品种(但不是所有)都是 在半单仿射代数群作用下的射影齐次。 该建议的主旨是研究下列类别的不可压缩性 品种(取决于素数p)： (A)特征为p的域的可分p-初等扩张的范数环面上的一般挠子。 在不同于p的特征中，结果已经成立。证据是有用的 关于系数在p元有限域上的Chow群的Steenrod运算。是这样的 在特征为p的域上不能进行运算。 (B)超曲面NRd=Const，其中NRd是p-素数中心单代数的约化范数。 这种方法是基于对品种平滑压实的动机的研究。 (C)与2-初等除代数有对合关系的酉Grmannians 整体型。我们希望得到与上一节解释的结果类似的结果。 关于辛对合。 所有上述问题都可以用某些代数圈的合理性来表示。 我的目的也是证明关于有理循环从函数域下降的结果 变种到基场(与R.Fino接合)。我们感兴趣的品种之一是 由p次中心除代数的约化范数给出的簇NRD=const(结果是 在特征0中已知)。 最后，我们计划研究连通K-理论中的运算，并可能应用到上面 有问题。

The main thrust of the proposal is to study numerical and discrete invariants of torsors and projective homogeneous varieties, such as essential and canonical dimensions, and their applications to the theory of algebraic groups over non-closed fields and related structures. The idea of building global information/objects out of compatible local data and the nature of the obstruction to doing so, are of great importance to mathematics and physics (manifolds are the quintessential example of a global object constructed by "patching together" local data). The algebraic version of manifolds (needed for example in applications to number theory, but also in far removed areas such as genetics) are called varieties, or more generally schemes. The theory of torsors provides the language and tools to measure the obstructions to patching local data together (say to construct a variety or scheme). The theory of essential dimension could be thought as a sort of "complexity theory" of torsors. Linked to this is the concept of canonical dimension and of compressible variety (a tool that allows us to know when certain type of constructions could not get simpler). Having introduced the central pieces of my research, I will proceed now to give a more detailed description of some of my current research and objectives. An algebraic variety is called incompressible, if all its rational endomorphisms are dominant. Results on incompressibility of varieties have numerous applications, especially to computation of essential dimension of algebraic structures. Incompressibility for several classes of varieties has been already established by the applicant. Most of the varieties (but not all of them) are projective homogeneous under an action of a semisimple affine algebraic group. The main thrust of the proposal is the study of incompressibility for the following classes of varieties (depending on a prime integer p): (a) Generic torsors over norm tori of separable p-primary extensions of fields of characteristic p. In characteristic different from p, the result has been already established. The proof makes use of Steenrod operations on Chow groups with coefficients in the finite field of p elements. Such operations are not available over fields of characteristic p. (b) Hypersurfaces Nrd = const, where Nrd is the reduced norm of a p-primary central simple algebra. The approach is based on study of motives of smooth compactifications of the variety. (c) Unitary grassmannians associated to a 2-primary division algebra endowed with an involution of unitary type. We expect to obtain an analogue of the results explained in the previous section concerning symplectic involutions. All of the above questions can be expressed in terms of the rationality of certain algebraic cycles. My objective is also to prove results on descent of rational cycles from the function field of a variety to the base field (joint with R. Fino). One of the varieties we are interested in is the variety Nrd = const given by the reduced norm of a degree p central division algebra (the result is already known in characteristic 0). Finally, we plan to study operations in connective K-theory with possible application to the above problems.