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

Siarhei Finski

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.

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

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.

Paper Structure

This paper contains 16 sections, 37 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Localization along the diagonal of the Bergman measures
  3. Case studies of off-diagonal behavior
  4. Localization in the Bernstein-Markov setting
  5. Reduction to measures supported away from subvarieties
  6. Decay of the kernel for measures supported away from a divisor
  7. On the weak limit of the diagonal Bergman measures
  8. Diagonal Bergman measures put no mass on subvarieties
  9. Forbidden region for the partial Bergman kernels
  10. Diagonal Bergman measures put no mass on closed pluripolar subsets
  11. Localization relative to weight variation
  12. Applications to the theory of Toeplitz operators and singular spaces
  13. Orthogonal polynomials and Toeplitz matrices
  14. Algebraic and spectral aspects of Toeplitz operators
  15. Spectral equidistribution criteria for Toeplitz operators
  16. ...and 1 more sections

Key Result

Theorem 1.1

For any Borel measure $\mu$ which does not give full mass to pluripolar subsets, the corresponding measures $\mu_k^{\rm{Berg}}$ do not asymptotically place mass away from the diagonal. In other words, for any compact subset $K \subset X \times X$, not intersecting the diagonal, we have

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 70 more