On the Localization of the Bergman Kernel and applications to Toeplitz theory

TL;DR

This paper investigates the localization properties of the Bergman kernel on complex manifolds with big line bundles, confirming a Zelditch conjecture and exploring implications for Toeplitz operator algebras and spectral distribution.

Contribution

It establishes the limiting behavior of the Bergman measure for Bernstein-Markov measures and demonstrates that Toeplitz operators form an algebra with equidistributed spectra.

Findings

Confirmed Zelditch's conjecture on Bergman measure localization

Proved Toeplitz operators form an algebra under composition

Showed spectral equidistribution for Toeplitz operators

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.