Abelian Varieties, Jacobians, and Applications

阿贝尔簇、雅可比行列式及其应用

• 批准号: 1053313
• 负责人: Samuel Grushevsky
• 金额: $ 13.16万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-05-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1053313&HistoricalAwards=false
• 关键词: Abelian; Varieties; Jacobians; Applications

The PI proposes to work with multiple collaborators to study topics in algebraic geometry, number theory, and string theory related to abelian varieties, curves, and their moduli. The PI will study the intersection homology of the moduli spaces of abelian varieties, investigate the failure of injectivity of the Torelli map for Prym varieties, and endeavor to prove by degeneration that the Schottky-Jung identities characterize Jacobians of curves among all abelian varieties. The PI will attempt to use the geometric properties of Jacobians to approach Coleman's conjecture on the finiteness of the number of Jacobians of a fixed large genus with complex multiplication. By using real-normalized meromorphic differentials, the PI will work on constructing complete subvarieties of the moduli space of curves. The PI will also investigate further mathematical and physical properties of superstring scattering amplitudes an ansatz for which he proposed.Algebraic curves (aka Riemann surfaces) are real two-dimensional surfaces with a metric on them. They arise in many areas of mathematics, and are also fundamental to string theory as worldsheets (trajectories) of strings propagating in space. One can associate to any algebraic curve its Jacobian - it is an algebraic variety (a geometric set of points, such that there is an operation of "adding" two points together), and knowing the Jacobian one can recover the curve uniquely. This project aims to utilize and further study the intricate interplay between the geometry of the curve and of its Jacobian in order to further our understanding of curves and abelian varieties. Progress made on the questions addressed by this project would have implications in mathematics and physics going beyond algebraic and complex geometry, particularly in number theory, integrable systems, partial differential equations, and perturbative string theory.

PI建议与多个合作者一起研究代数几何，数论和与阿贝尔簇，曲线及其模相关的弦理论。PI将研究交换簇的模空间的交同调，研究Prym簇的Torelli映射的内射性的失败，并奋进通过退化证明Schottky-Jung恒等式表征所有交换簇中曲线的Jacobian。PI将尝试使用雅可比矩阵的几何性质来接近科尔曼关于具有复数乘法的固定大亏格的雅可比矩阵的数量的有限性的猜想。通过使用实规格化亚纯微分，PI将致力于构造曲线模空间的完全子簇。PI还将进一步研究超弦散射振幅的数学和物理性质，他提出了一种新的方法。代数曲线（又名黎曼曲面）是真实的二维曲面，其上有一个度量。它们出现在数学的许多领域，也是弦理论的基础，因为它们是弦在空间中传播的世界面（轨迹）。人们可以将其雅可比矩阵与任何代数曲线相关联-它是代数簇（点的几何集合，使得存在将两个点“相加”在一起的操作），并且知道雅可比矩阵可以唯一地恢复曲线。这个项目的目的是利用和进一步研究曲线的几何形状和它的雅可比矩阵之间的复杂的相互作用，以进一步我们的曲线和阿贝尔品种的理解。在这个项目所解决的问题上所取得的进展将对数学和物理学产生影响，超越代数和复几何，特别是在数论，可积系统，偏微分方程和微扰弦理论。