Upper Bounds for Bergman Kernels Associated to Positive Line Bundles with Smooth Hermitian Metrics

Abstract

Off-diagonal upper bounds are established for Bergman kernels associated to powers \(L^\lambda \) of holomorphic line bundles L over compact complex manifolds, asymptotically as the power \(\lambda \) tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is \(C^\infty \) but not necessarily real analytic. The bounds are of the form \(\exp (-h(\lambda )\sqrt{\lambda \log \lambda })\) where h tends to infinity at a non-universal rate. This form is best possible.