Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

TL;DR

This paper establishes exponential decay bounds for Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded below, extending previous results to non-Kähler and unbounded geometry cases.

Contribution

It provides explicit geometric conditions ensuring coercivity of the complex Laplacian, leading to decay bounds without assuming Kähler or bounded geometry.

Findings

Proves Agmon-type bounds for Bergman kernels under general conditions.

Extends known bounds to non-Kähler, unbounded geometry manifolds.

Uses localization formulas and heat equation inequalities in the proof.

Abstract

Given a smooth positive measure $\mu$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate version of the Bochner--Kodaira--Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be K\"ahler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schr\"odinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the so-called "twisted basic identities" are standard basic identities with respect to conformally K\"ahler metrics.