Pluripotential Theory and Random Geometry on Compact Complex Manifolds

该项目涉及复分析、复几何和势理论等数学领域。复分析研究依赖于复数变量的函数。复杂分析和势理论为解决纯数学和应用数学（如图像和信号处理）和物理学（如量子力学和统计物理）的其他领域的重要问题提供了强大的工具。该项目将侧重于促进这些领域的知识和理解的各种问题集合。复分析和势理论的新技术将应用于复杂几何和代数几何、数学物理和数论等领域的问题。例如，该项目将研究全纯线束的截面和相关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。该项目将通过有效地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。该项目还将有助于组织若干复杂变数的会议。这些活动将汇集知名数学家、早期职业研究人员和研究生，讨论数学研究和学生指导。该项目将在紧复流形的背景下，解决源自多能势理论和随机复几何领域的问题。其中一些问题对复杂几何和代数几何、数学物理或数论都有重要的应用。一个统一的主题是关注多次谐波函数和正闭合电流作为研究对象或工具。第一个研究方向涉及紧复空间上的量化问题。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。对于紧复空间上的奇异厄密全纯线束序列，存在着平方可积全纯截面的自然Bergman空间。在这些空间的维数增长、Fubini-Study流的收敛性以及相关Bergman核函数的渐近性方面，将考虑适当的曲率正性假设。另一个要考虑的主题是Bergman空间中m元组的随机序列的公共零的渐近分布，其中将特别注意收敛速度的估计。对于沿解析子集消失到高阶的全纯截面，我们将研究相应的部分Bergman核的渐近性。另一个研究方向是紧化Kaehler流形的多势理论。这里出现了与当地环境不同的有趣的新现象。研究者将研究复Monge-Ampere算符定义良好的拟多次谐波函数的最大域，以及相应的拟多次谐波Green函数的奇异性。定义在解析子变种上的拟多次调和函数的扩展和正则化将发挥作用。最后，本课题将探讨投影流形上任意二维正闭流的Lelong数的上能级集的几何性质，并将阐明与上同调的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.