Hodge theoretic and algebraic approaches to the theory of motives

提出的研究涉及动机理论和代数循环理论中的两个主题。第一个是霍奇理论。受Mark Green和Phillip Griffiths关于Hodge猜想的工作以及Richard Hain和David Reed关于代数循环的工作的启发，PI和Gregory Pearlstein定义了与光滑投影复变中的Hodge类相关的度量线束序列，称为双扩展线束。所提出的研究的主要目标是了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霍奇猜想。第二部分是关于代数群上同调不变量的研究。这些不变量与代数群G在域F上的任意量相关，域F是F的伽罗瓦上同调中的一类，尽管它们看起来很难明确地计算，但上同调不变量是非常自然的对象，人们希望它们能给出一个代数群的量的全部信息。通过Burt Totaro的观察，利用G的分类空间的动机上同调，可以计算群G的上同调不变量。PI打算利用Totaro的观察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，激发了第一个提出的主题，问是否上同调决定代数循环。同样，第二个提出的主题是问在多大程度上上同调不变量决定torsor。由于线性不变量通常比非线性不变量更容易处理，因此这两个主题在代数几何和相关学科中都具有重要的基础意义。

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.