This research project in Mathematics by Dan Coman is in the areas of complex analysis, complex geometry and potential theory. These areas are central to modern mathematics and 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; statistical physics).

A major goal of the project was concerned with quantization problems on complex manifolds and with the asymptotic distribution of zeros of random holomorphic sections. Given a line bundle L endowed with a Hermitian metric h on a complex manifold X, one considers the Bergman spaces of square-integrable holomorphic sections of the tensor powers of L defined naturally using the metric data. Associated with these spaces are the Bergman kernel functions and the Fubini-Study currents, and one studies the asymptotic expansion of the first and the convergence of the latter. As application to physics, these are related to the quantum mechanics of electrons in magnetic field. In recent years, many authors considered the case when the metric h is smooth. Together with George Marinescu, we initiated a program to study these objects in the more general case when h is singular. By improving existing methods as well as developing new techniques, we were able to generalize results in random complex geometry in two directions: we allowed the base space X to be singular, so a complex space rather than a manifold, and we considered arbitrary sequences of singular Hermitian holomorphic line bundles on X instead of the sequence of powers of a fixed bundle L (the results are published in joint papers with Bayraktar, Ma, Marinescu). In joint work with Marinescu and Nguyen we studied partial Bergman kernels corresponding to spaces of holomorphic sections vanishing to high orders along subvarieties of X.

Another goal of the project was concerned with problems from pluripotential theory on a compact Kaehler manifold. Lelong numbers are important biholomorphic invariants which measure the singularities of positive closed currents. Given such currents on multiprojective spaces, in joint work with my former student Heffers we gave geometric descriptions of the set of points where the Lelong numbers exceed certain constants which depend on dimension and cohomology. In joint work with Guedj, Sahin and Zeriahi, we study properties of the complex Monge-Ampere operator and related classes of quasiplurisubharmonic functions in the setting of compact toric manifolds.

During the award period, six papers were published and three more were finalized and submitted for publication. I was invited to participate to seven national and international conferences and workshops, and I gave 11 invited talks about the results obtained as outcomes of this project.

The project had an impact on the development of human resources. A graduate student, James Heffers, worked on his dissertation under my supervision on topics related to this project, and defended his thesis in April 2018. He received summer support and travel support from this grant. After graduation, James Heffers accepted an Assistant Professor position at the University of Michigan, Ann Arbor, for the period August 2018-May 2022. I have been organizing conferences in several complex variables and complex geometry, bringing together established mathematicians as well as young researchers and graduate students, to discuss Mathematics and their teaching.

Last Modified: 07/01/2021
Modified by: Dan Coman