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

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.