Algebraic Cycles, Hodge Theory and Arithmetic

代数几何的古老起源在于对多项式方程解集的研究，其中复几何、辛几何和算术几何紧密地联系在一起。该学科中许多最引人注目的最新发展都是通过对这些方面的综合考虑而发生的：例如，辛几何和复几何之间的对偶性，即镜像对称；以及围绕Beilinson， Bloch和Hodge关于广义代数循环的超越不变量的猜想的工作。在之前的工作中，提出者已经发现了迄今为止未知的所谓的Abel-Jacobi不变量的具体公式，并应用它们来解释局部镜像对称中瞬时数的渐近行为，以及证明关于循环本身的新结果。本项目将考虑广义循环及其不变量在Hodge理论、弦理论和算术代数几何中密切相关的问题中的新应用。它还将解决一些鲜为人知的时期地图和这些应用程序背后的时期域的问题。具体来说，我们计划研究由Abel-Jacobi不变量及其极限结合在一起的几个主题：代数变量的Hodge理论模的边界行为（在周期域和极限混合Hodge结构的背景下）；循环在无理性证明中的应用以及广义环在同调镜像对称和杂杂/ II型弦对偶中的作用，以期建立作为理论物理基础新工具的高代数k理论。我们期望，反过来，这些物理应用将揭示算术和辛几何之间的神秘联系，这是局部镜像对称结果所突出的。

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.