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.

该项目涉及复分析、复几何和势理论的数学领域。复分析研究依赖于复数变量的函数。复分析和势理论为解决来自纯数学和应用数学其他领域的重要问题提供了强有力的工具（例如，图像和信号处理）和物理学（例如，量子力学和统计物理学）。该项目将侧重于一系列不同的问题，以增进这些领域的知识和理解。从复杂的分析和潜在的理论的新技术将被应用到起源于复杂和代数几何，数学物理和数论等领域的问题。例如，该项目将研究全纯线丛的截面和相关的Bergman核函数的渐近性。例如，这些主题与磁场中粒子的量子力学有关。该项目将通过有效整合研究和教育来影响人力资源的开发，并将包括对博士论文的监督。该项目还将有助于在几个复杂的变量中组织会议。这些活动将汇集建立数学家，早期职业研究人员和研究生讨论数学研究和学生辅导。这个项目将解决起源于多能理论和随机复几何领域的问题，在紧凑的复杂流形的设置。其中一些问题在复几何和代数几何、数学物理或数论中有重要的应用。一个统一的主题是一个专注于pluisubharmonic函数和积极的封闭电流作为调查对象或作为工具。第一个研究方向涉及紧复空间上的量子化问题。这些问题在统计物理学（通过量子混沌）和数论（通过模形式的量子唯一遍历性）中都有应用。与紧复空间上的奇异Hermitian全纯线丛序列相联系，存在平方可积全纯截面的自然Bergman空间。适当的积极性假设曲率将被认为与这些空间的尺寸的增长，收敛的Fubini研究电流，以及相关的伯格曼核函数的渐近性。另一个要考虑的问题是伯格曼空间中m元组的随机序列的公共零点的渐近分布，其中特别注意收敛速度的估计。关于沿沿着解析子集消失到高阶的全纯部分，将研究相应的部分Bergman核的渐近性。另一个研究方向涉及紧致Kaehler流形上的多势理论。这里出现了有趣的新现象，与当地环境不同。调查员将研究quasipulisubharmonic函数的最大域上的复杂的蒙格-安培运营商是很好地定义，和相应的quasipulisubharmonic绿色功能的奇异性。定义在解析子簇上的拟复次调和函数的扩张和正则化将发挥作用。最后，该项目将探索投影流形上任意二维正闭合流的Lelong数的上层集合的几何性质，并将阐明与上同调的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。