Global motivic integration and the cohomology of moduli spaces

全局动机积分和模空间的上同调

• 批准号: 327639-2006
• 负责人: Dhillon, Ajneet
• 金额: $ 0.66万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=397887
• 关键词: Global; motivic; integration; cohomology; moduli

Algebraic geometry, at the crudest level, is the study of the geometry of solutions of polynomial equations in many variables. In order to proceed one must enlarge the collection of solution spaces to a larger collection consisting of spaces that are algebraic gluings of such spaces. We will refer to such spaces as varieties. In the 1950's the Grothendieck school taught us that it is also important to remember the defining equations. The geometry of such solution spaces is usually studied by associating to the space an algebraic invariant. In 1995, inspired by ideas from calculus and number theory, M. Kontsevich proved some conjectures pertaining to some invariants associated to a special class of varieities. The theory of motivic integration was born. The theory has been developed over the last ten years most notably by J. Denef and  F. Loeser. The aim of this research is to further extend this theory. The second part of this proposal aims to study some invariants of the a special variety, the so called moduli space of semistable vector bundles over an algebraic curve.

从最粗略的层面来看，代数几何是对多变量多项式方程解的几何的研究。为了继续下去，必须将解空间的集合扩大到一个更大的集合，该集合由这些空间的代数粘合的空间组成。我们将此类空间称为品种。 20 世纪 50 年代，格洛腾迪克学派告诉我们，记住定义方程也很重要。通常通过将代数不变量与空间相关联来研究此类解空间的几何形状。 1995 年，受到微积分和数论思想的启发，M. Kontsevich 证明了一些与一类特殊变体相关的不变量的猜想。动机整合理论由此诞生。该理论在过去十年中得到了发展，最著名的是 J. Denef 和 F. Loeser。本研究的目的是进一步扩展这一理论。该提案的第二部分旨在研究特殊类型的一些不变量，即代数曲线上的半稳定向量丛的所谓模空间。