Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels

TL;DR

This paper shows that weight operators from submultiplicative filtrations on complex manifolds are Toeplitz operators and studies their Bergman kernel asymptotics, refining convergence results of jumping measures.

Contribution

It establishes the Toeplitz operator nature of weight operators and provides a detailed analysis of weighted Bergman kernel asymptotics in this context.

Findings

Weight operators are Toeplitz operators.

Asymptotic behavior of weighted Bergman kernels is characterized.

Refined convergence of jumping measures towards pushforward measures.

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.