Hodge theoretic and algebraic approaches to the theory of motives

动机理论的霍奇理论和代数方法

• 批准号: 1103269
• 负责人: Patrick Brosnan
• 金额: $ 16万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103269&HistoricalAwards=false
• 关键词: Hodge; theoretic; algebraic; approaches; theory

The proposed research concerns two topics within the theory of motives and algebraic cycles. The first is Hodge theory. Motivated by work of Mark Green and Phillip Griffiths on the Hodge conjecture and by work of Richard Hain and David Reed on algebraic cycles, the PI and Gregory Pearlstein have defined a sequence of metrized line bundles called biextension line bundles associated to Hodge classes in smooth,projective complex varieties. The main goal of the proposed research is to understand the metrics and the asymptotics of the metric at infinity in the hope of gaining insight into the geometry of moduli spaces and into the Hodge conjecture.The second part of the proposed research concerns cohomological invariants associated to algebraic groups. These are invariants associating to any torsor for an algebra group G over a field F a class in the Galois cohomology of F. Although they seem difficult to compute explicitly, cohomological invariants are very natural objects, and one would hope that they give full information about the torsors for an algebraic group. By an observation of Burt Totaro, the cohomological invariants of a group G are computatable in terms of the motivic cohomology of the classifying space of G. The PI intends to use Totaro's observation to compute cohomological invariants of the spinor group and related groups.The unifying theme in both proposed topics is to understand to what extent problems in algebraic geometry can be linearized and studied using cohomology. The Hodge conjecture, which motivates the first proposed topic, asks if cohomology determines algebraic cycles. Similarly, the second proposed topic asks to what extent cohomological invariants determine torsors. Since linear invariants are usually more tractable than non-linear ones, both topics are of fundamental importance in algebraic geometry and related subjects.

拟议的研究涉及两个主题内的理论动机和代数圈。 第一个是Hodge理论。 受到马克·绿色和菲利普·格里菲思关于霍奇猜想的工作以及理查德·海恩和大卫·里德关于代数循环的工作的启发，PI和格雷戈里·珀尔斯坦定义了一系列度量化的线丛，称为双延线丛，与光滑的射影复簇中的霍奇类相关。 本研究的主要目标是理解度量和度量在无穷远处的渐近性，以期深入了解模空间的几何和Hodge猜想。本研究的第二部分涉及与代数群相关的上同调不变量。 这些不变量与域F上的代数群G的任何torsor相关联，是F的伽罗瓦上同调中的一类。 虽然上同调不变量似乎很难明确计算，但它们是非常自然的对象，人们希望它们能给出代数群的torsors的全部信息。 根据Burt Totaro的一个观察，群G的上同调不变量可以用G的分类空间的动机上同调来计算。 PI打算使用Totaro的观察来计算旋量群和相关群的上同调不变量。这两个主题的统一主题是理解代数几何中的问题在多大程度上可以线性化并使用上同调进行研究。 霍奇猜想，这激发了第一个提出的主题，问如果上同调确定代数循环。同样，第二个提出的主题问到什么程度上同调不变量确定torsors。 由于线性不变量通常比非线性不变量更易处理，因此这两个主题在代数几何和相关学科中具有根本的重要性。