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Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels

Siarhei Finski

Abstract

We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray.

Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels

Abstract

We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray.

Paper Structure

This paper contains 14 sections, 29 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. Acknowledgement.
  4. The spaces of norms, Kähler forms and geodesic rays
  5. Geodesics in the space of Hermitian norms.
  6. Mabuchi geodesics.
  7. Test configurations and submultiplicative filtrations.
  8. Construction of geodesic rays in Mabuchi space.
  9. Example 1.
  10. Example 2.
  11. Asymptotics of weight operators, proofs of Theorems \ref{['thm_main1']}, \ref{['thm_main2']}
  12. Transfer operator between $L^2$-norms, a proof of Theorem \ref{['thm_trasnfer']}
  13. Functional calculus for Toeplitz operators
  14. Superadditivity of weighted Bergman kernels, a proof of Theorem \ref{['thm_berg_conv']}

Key Result

Theorem 1.1

For a bounded submultiplicative filtration $\mathcal{F}$ on $R(X, L)$, the sequence of functions $x \mapsto \frac{1}{k^n} B_k^{\mathcal{F}, g}(x)$, $x \in X$, $k \in \mathbb{N}$, is uniformly bounded and converges pointwise to a function which equals $g(\phi(h^L, \mathcal{F}))$ almost everywhere. If

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1: cf. InterpSp and FinTits
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • ...and 47 more