Pluripotential Theory and Random Geometry on Compact Complex Manifolds

Syracuse University, Syracuse NY

Investigators

Abstract

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.