Sharp $C^{1,\bar1}$ estimates in K\"ahler quantization and non-pluripolar Radon measures

TL;DR

This paper establishes sharp $C^{1,ar1}$ estimates for the Bergman kernel in K"ahler geometry, providing bounds based on the potential function and applying these results to quantization of K"ahler currents and Radon measures.

Contribution

It introduces optimal regularity estimates for the Bergman kernel in K"ahler quantization, extending the understanding of quantization for non-pluripolar Radon measures.

Findings

Derived sharp $C^{1,ar1}$ bounds for the Bergman kernel and metric.

Achieved optimal $C^{1,ar1}$-convergence in K"ahler quantization.

Proved that any non-pluripolar Radon measure admits a quantization.

Abstract

Let $K_\varphi$ denote the weighted Bergman kernel associated to a plurisubharmonic function $\varphi$. We obtain upper bounds and positive lower bounds for the Bergman metric $i\partial \bar{\partial} \log K_\varphi$, expressed solely in terms of upper bounds and positive lower bounds of $i\partial \bar{\partial}\varphi$. Our approach applies in both local and compact K\"ahler settings. As an immediate application we obtain the optimal $C^{1,\alpha}$-convergence for the quantization of K\"ahler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact K\"ahler manifold admits a quantization.