On the Localization of the Bergman Kernel and applications to Toeplitz theory
Complex Variables
2026-03-25 v2 Algebraic Geometry
Classical Analysis and ODEs
Functional Analysis
Abstract
For a compact complex manifold endowed with a big line bundle and a Radon measure, we study the localization phenomena of the associated Bergman (or Christoffel-Darboux) kernel. For Bernstein-Markov measures, this results in the determination of the limiting off-diagonal Bergman measure, thereby confirming a conjecture of Zelditch. We then turn to applications in the theory of Toeplitz operators, showing in particular that they form an algebra under composition. Building on this, we then show that for Bernstein-Markov measures, the spectrum of Toeplitz operators equidistributes.
Keywords
Cite
@article{arxiv.2508.03416,
title = {On the Localization of the Bergman Kernel and applications to Toeplitz theory},
author = {Siarhei Finski},
journal= {arXiv preprint arXiv:2508.03416},
year = {2026}
}
Comments
41 pages; final version which subsumes arXiv:2506.01610