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

Zbigniew Błocki, Tamás Darvas

TL;DR

This work develops sharp on-diagonal $C^{1,\bar{1}}$ estimates for Bergman metrics in Kähler quantization, connecting $i\partial\bar{\partial}\log K^k_\varphi$ to bounds on $i\partial\bar{\partial}\varphi$ and proving $C^{1,\alpha}$ convergence of the quantized potentials $P^k_\varphi$ to $\varphi$ under degenerate but controlled curvature. It leverages both local and global (Kähler) Bergman kernel comparisons, together with Berndtsson’s quantized comparison principle, to derive explicit bounds for local and global Bergman kernels and their derivatives. The authors extend the quantization program beyond smooth or continuous potentials to full mass and finite-energy settings, proving weak convergence of quantum measures $M^k_\varphi$ to the Monge–Ampère measure $\omega_\varphi^n$ and, crucially, to non-pluripolar Radon measures via a broad envelope/perturbation framework. They also develop the Monge–Ampère energy quantization, showing that $I_k(H^k_\varphi)$ converges to $I(\varphi)$ and that perturbations by smooth functions are reflected at the quantum level, linking variational energy principles across the quantum-classical divide. Overall, the results significantly broaden the scope of Kähler quantization to degenerate and singular settings with concrete, optimal estimates and convergence statements.

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 Kähler settings. As an immediate application we obtain the optimal $C^{1,α}$-convergence for the quantization of Kähler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact Kähler manifold admits a quantization.

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

TL;DR

This work develops sharp on-diagonal estimates for Bergman metrics in Kähler quantization, connecting to bounds on and proving convergence of the quantized potentials to under degenerate but controlled curvature. It leverages both local and global (Kähler) Bergman kernel comparisons, together with Berndtsson’s quantized comparison principle, to derive explicit bounds for local and global Bergman kernels and their derivatives. The authors extend the quantization program beyond smooth or continuous potentials to full mass and finite-energy settings, proving weak convergence of quantum measures to the Monge–Ampère measure and, crucially, to non-pluripolar Radon measures via a broad envelope/perturbation framework. They also develop the Monge–Ampère energy quantization, showing that converges to and that perturbations by smooth functions are reflected at the quantum level, linking variational energy principles across the quantum-classical divide. Overall, the results significantly broaden the scope of Kähler quantization to degenerate and singular settings with concrete, optimal estimates and convergence statements.

Abstract

Let denote the weighted Bergman kernel associated to a plurisubharmonic function . We obtain upper bounds and positive lower bounds for the Bergman metric , expressed solely in terms of upper bounds and positive lower bounds of . Our approach applies in both local and compact Kähler settings. As an immediate application we obtain the optimal -convergence for the quantization of Kähler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact Kähler manifold admits a quantization.
Paper Structure (14 sections, 20 theorems, 101 equations)

This paper contains 14 sections, 20 theorems, 101 equations.

Table of Contents

  1. Main results
  2. The $C^{1,\bar{1}}$ estimates for Kähler quantization.
  3. The $C^{1,\bar{1}}$ estimates for the local Bergman potential.
  4. Kähler quantization of non-pluripolar Radon measures.
  5. Acknowledgments.
  6. Preliminaries
  7. Comparison of local Bergman kernels.
  8. Comparison of the local and Kähler Bergman kernels.
  9. Berndtsson's quantized comparison principle
  10. The $C^{1,\bar{1}}$ estimates of Bergman potentials
  11. The local case.
  12. The compact Kähler case.
  13. Quantization of the Monge--Ampère energy
  14. Quantization of Radon measures

Key Result

Theorem 1.1

Suppose that $\varphi \in \mathcal{H}^{1,\bar{1}}_\omega$ satisfies $0 < a \omega \leq \omega_\varphi \leq A \omega.$ Then there exists a positive constant $C=C(X,\omega)$ such that In particular, $P^k_\varphi \to_{C^{1,\alpha}} \varphi$ for any $\alpha \in (0,1)$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 26 more