Hodge theoretic and algebraic approaches to the theory of motives

拟议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定义了一系列度量化线丛，称为双扩张线丛，它们与光滑的射影复变簇中的Hodge类有关。研究的主要目的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Hodge猜想。第二部分研究与代数群有关的上同调不变量。上同调不变量是与域F上代数群G的任意扭量相关的不变量，是F的Galois上同调中的一类。虽然上同调不变量看起来很难显式计算，但上同调不变量是非常自然的对象，人们希望它们能给出关于代数群的扭量的全部信息。通过Burt Totaro的观察，群G的上同调不变量可以根据G的分类空间的基元上同调来计算。PI打算利用Totaro的观察来计算旋量群及其相关群的上同调不变量。这两个主题的统一主题是了解代数几何中的问题在多大程度上可以线性化并利用上同调来研究。霍奇猜想激发了第一个提出的主题，它问上同调是否决定了代数圈。类似地，第二个提出的主题是问上同调不变量在多大程度上决定了扭量。由于线性不变量通常比非线性不变量更容易处理，这两个主题在代数几何和相关学科中都是基本重要的。

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.