Isotropic motives and affine quadrics

本提案的主题领域是由主要研究者引入的代数变量的新动机不变量。这些方法加强和扩展了V.Voevodsky (Fields Medal 2002)， M.Levine， A.Merkurjev， F.Morel， M.Rost, a. suslin等人在代数领域的突破性发展，提出的研究处于代数几何、拓扑和代数的边界。这些领域使用不同的工具集，它们之间的任何联系都允许从不同的角度研究相同的数学对象。拓扑学是几何的“柔性版本”，最简单的拓扑对象是一个点。在代数几何中也有点，但这些点取决于场的选择，因此在形状上有很大的不同。这是代数几何世界丰富的源泉。事实上，正如Morel-Voevodsky的惊人工作所证明的那样，拓扑世界只是它的“玩具版本”。拓扑对象的基本信息包含在其同调中，而代数变量的同调信息则编码在其动机中。这种动机的理论是由沃沃茨基提出的。首席研究员介绍了一种基于所谓的“各向同性实现函子”的动机研究新方法，该方法为动机分配了一系列“阴影”。这些“阴影”是由地面场的扩展参数化的（或者，如果你愿意，所有的代数几何点），并且在复杂性上与“拓扑动机”（奇异复核）相似。因此，一个复杂的对象被一组简单对象所取代。本文的主要目的是研究这些函子，并将它们推广为完全动机不变量。建立有限系数和有理系数代数环的数值等价的联系。将它们应用于二次曲面和Voevodsky范畴的Picard群的不变量的计算，以及代数循环上的Rost幂零猜想和标准猜想。第二个目的是将这些不变量推广到同伦上下文中，并研究经典拓扑对象的“各向同性”版本。这项研究将在诺丁汉大学数学科学学院进行。

The subject area of this proposal is the new motivic invariants of algebraic varieties introduced by the principal investigator. These methods enhance and extend the groundbreaking development in algebra associated with the names of V.Voevodsky (Fields Medal 2002), M.Levine, A.Merkurjev, F.Morel, M.Rost, A.Suslin.The proposed research lies at the boundary of algebraic geometry, topology and algebra. These areas use different sets of tools, and any connection between them permits to study the same mathematical object from different perspectives. Topology is a "flexible version" of geometry, and the simplest topological object is a point. In algebraic geometry there are also points, but these depend on the choice of a field, and so, substantially vary in shape. This is the source of richness of the algebro-geometric world. In fact, as was demonstrated by the spectacular work of Morel-Voevodsky, the topological world is just a "toy version" of it. The basic information on a topological object is contained in its homology, while the homological information on an algebraic variety is encoded in its motive. The theory of such motives was developed by Voevodsky.The principal investigator has introduced the new approach to the study of motives based on the, so called, "isotropic realization functors", which assign to a motive a family of its "shadows". These "shadows" are parameterized by the extensions of the ground field (or, all the algebro-geometric points, if you want) and are similar in complexity to "topological motives" (singular complexes). Thus, a complicated object is substituted by an array of simple ones.The principal aim of the proposal is to study these functors and extend them to a complete motivic invariant. To establish the connection to the numerical equivalence of algebraic cycles with finite and rational coefficients. To apply them to the computation of invariants of quadrics and the Picard group of the Voevodsky category, as well as to the Rost Nilpotence Conjecture and the Standard Conjectures on algebraic cycles. The second aim of the proposal is to generalize these invariants to the homotopic context, and study "isotropic" versions of classical topological objects.The research will be undertaken at the School of Mathematical Sciences, University of Nottingham.