Pluripotential Theory and Random Geometry on Compact Complex Manifolds

紧复流形上的多势理论和随机几何

• 批准号: 2154273
• 负责人: Dan Coman
• 金额: $ 23.71万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154273&HistoricalAwards=false
• 关键词: Pluripotential; Theory; Random; Geometry; Compact

This project lies in the mathematical fields of complex analysis, complex geometry, and potential theory. Complex analysis studies functions depending on variables that are complex numbers. Complex analysis and potential theory provide powerful tools for solving important problems from other fields of pure and applied mathematics (e.g., image and signal processing) and physics (e.g., quantum mechanics and statistical physics). The project will focus on a diverse collection of questions advancing knowledge and understanding in these fields. New techniques from complex analysis and potential theory will be applied to questions originating in fields as diverse as complex and algebraic geometry, mathematical physics, and number theory. For example, the project will investigate sections of holomorphic line bundles and the asymptotics of the related Bergman kernel functions. These topics are related, for instance, to the quantum mechanics of particles in magnetic fields. The project will impact the development of human resources by effectively integrating research and education, and will include the supervision of doctoral theses. The project will also contribute to the organization of conferences in several complex variables. These events will bring together established mathematicians, early-career researchers, and graduate students to discuss mathematics research and student mentoring. This project will address questions originating in the fields of pluripotential theory and random complex geometry, in the setting of compact complex manifolds. Some of these questions have important applications to complex and algebraic geometry, mathematical physics, or number theory. A unifying theme is a focus on plurisubharmonic functions and positive closed currents as objects of investigation or as tools to be employed. The first direction of research involves quantization problems on compact complex spaces. Such questions have applications to both statistical physics (via quantum chaos) and number theory (via quantum unique ergodicity for modular forms). Associated to a sequence of singular Hermitian holomorphic line bundles over a compact complex space, there are natural Bergman spaces of square-integrable holomorphic sections. Suitable positivity assumptions on curvature will be considered in connection with the growth of the dimension of these spaces, the convergence of the Fubini-Study currents, and the asymptotics of the associated Bergman kernel functions. Another topic to be considered is the asymptotic distribution of common zeros of random sequences of m-tuples of sections in the Bergman spaces, where special attention will be paid to estimates for the speed of convergence. In connection with holomorphic sections that vanish to high order along an analytic subset, the asymptotics of the corresponding partial Bergman kernels will be studied. Another direction of research deals with pluripotential theory on compact Kaehler manifolds. Here interesting new phenomena arise, distinct from the local setting. The investigator will study the largest domain of quasiplurisubharmonic functions on which the complex Monge-Ampere operator is well defined, and singularities of the corresponding quasiplurisubharmonic Green functions. Extension and regularization of quasiplurisubharmonic functions defined on analytic subvarieties will play a role. Finally, the project will explore geometric properties of upper-level sets of Lelong numbers of positive closed currents of arbitrary bidimension on projective manifolds, and will elucidate connections to cohomology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目在于复杂分析，复杂几何学和潜在理论的数学领域。复杂的分析研究的功能取决于复数的变量。复杂的分析和潜在理论提供了从其他纯数学和应用数学（例如，图像和信号处理）和物理（例如量子力学和统计物理学）中解决重要问题的强大工具。该项目将重点关注各种各样的问题，这些问题推进这些领域的知识和理解。复杂分析和潜在理论的新技术将应用于源于复杂和代数几何，数学物理学和数字理论等领域的问题。例如，该项目将研究霍明型线束的部分和相关的伯格曼内核函数的渐近学。这些主题与磁场中颗粒的量子力学有关。该项目将通过有效整合研究和教育来影响人力资源的发展，并将包括博士论文的监督。该项目还将有助于几个复杂变量的会议组织。这些事件将汇集成熟的数学家，早期研究人员和研究生，讨论数学研究和学生指导。该项目将在紧凑的复杂歧管的环境中解决源自多电位理论和随机复杂几何学领域的问题。其中一些问题在复杂和代数的几何形状，数学物理学或数字理论中具有重要的应用。一个统一的主题是专注于plurisubharmonic功能和积极的封闭电流作为调查对象或要使用的工具。研究的第一个方向涉及紧凑的复杂空间上的量化问题。此类问题在统计物理（通过量子混乱）和数字理论（通过模块化形式的量子独特性）都有应用。与紧凑的复杂空间上的一系列奇异隐居全态线束相关，有天然的伯格曼空间，可容纳正方形的全体形态切片。将考虑与这些空间维度的生长，fubini-study电流的收敛性以及相关伯格曼内核函数的渐近性有关的合适阳性假设。要考虑的另一个主题是伯格曼空间中M-Tubles的随机序列的常见零分布，在这里将特别注意估计收敛速度。与沿分析子集消失至高阶的尸体形态部分有关，将研究相应的部分伯格曼核的渐近学。研究的另一个方向涉及有关紧凑型Kaehler歧管的多能理论。这里出现了有趣的新现象，与当地环境不同。研究者将研究复杂的Monge-Ampere操作员在Quasiplurisubharmonic功能的最大领域，并在其上定义了相应的quasiplurisubharmonic绿色功能。在分析亚地区定义的Quasiplurisubharmonic功能的扩展和正规化将发挥作用。最后，该项目将探索在投影歧管上任意二维的正面封闭电流的上层级别集合的几何特性，并将阐明与共同体的联系。该奖项反映了NSF的法定任务，并通过该基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的支持。