Algebraic Cycles, Hodge Theory and Arithmetic

代数圈、霍奇理论和算术

• 批准号: EP/H021159/1
• 负责人: Matthew Kerr
• 金额: $ 13.01万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH021159%2F1
• 关键词: Algebraic; Cycles; Hodge; Theory; Arithmetic

The antique origins of algebraic geometry lie in the study of solution sets of polynomial equations, in which complex, symplectic, and arithmetic geometry are bound tightly together. Many of the most spectacular recent developments in the subject have occurred through the consideration of these aspects in tandem: for example, the duality between symplectic and complex geometry that is mirror symmetry; and the body of work surrounding conjectures of Beilinson, Bloch, and Hodge on the transcendental invariants of generalized algebraic cycles. In previous work, the proposer has found hitherto unknown concrete formulas for the so-called Abel-Jacobi invariants and applied them to explain asymptotic behavior of instanton numbers in local mirror symmetry, as well as to prove new results on cycles themselves.This project will consider novel applications of generalized cycles and their invariants to closely related problems in Hodge theory, string theory and arithmetic algebraic geometry. It will also work out poorly understood aspects of period maps and period domains underlying some of these applications.Specifically, we plan to study several topics which are bound together by Abel-Jacobi invariants and their limits: the boundary behavior of Hodge-theoretic moduli of algebraic varieties (in the context of period domains and limit mixed Hodge structures); applications of cycles to irrationality proofs; and the role played by generalized cycles in homological mirror symmetry and heterotic/type II string duality, with a view to establishing higher algebraic K-theory as a fundamental new tool in theoretical physics. We expect that, in turn, these physics applications will shed light on the mysterious connection between arithmetic and symplectic geometry highlighted by the local mirror symmetry result.

代数几何形状的古董起源在于对多项式方程溶液集的研究，其中复杂，符号和算术几何形状紧密地结合在一起。该受试者的许多最近期发展是通过同时考虑这些方面的：例如，是镜像对称性的符号和复杂几何形状之间的二元性；围绕贝林森，布洛赫和霍奇的猜想的工作体系，对广义代数周期的先验不变。 In previous work, the proposer has found hitherto unknown concrete formulas for the so-called Abel-Jacobi invariants and applied them to explain asymptotic behavior of instanton numbers in local mirror symmetry, as well as to prove new results on cycles themselves.This project will consider novel applications of generalized cycles and their invariants to closely related problems in Hodge theory, string theory and arithmetic代数几何形状。它还将解决这些应用中某些应用程序基础的周期地图和周期域的知识不足。特别是，我们计划研究几个主题，这些主题由Abel-Jacobi不变式绑定在一起及其限制及其限制：Hodge Theoretic Theoretic Moduli的边界行为（在时期域和周期域和限制混合杂种结构的背景下）;周期在非理性证明中的应用；以及广义周期在同源镜对称性和异形/II型字符串二元性中所起的作用，以期将较高的代数K理论作为理论物理学中的基本新工具。我们预计，这些物理应用程序又将阐明局部镜像对称性结果突出显示算术和象征性几何形状之间的神秘联系。