Peak sections and Bergman kernels on Kähler manifolds with complex hyperbolic cusps

By revisiting Tian’s peak section method, we obtain a localization principle of the Bergman kernels on Kähler manifolds with complex hyperbolic cusps, which is a generalization of Auvray-Ma-Marinescu’s localization result Bergman kernels on punctured Riemann surfaces [Auvray-Ma-Marinescu, Math. Ann., 2021]. Then we give some further estimates when the metric on the complex hyperbolic cusp is a Kähler-Einstein metric or when the manifold is a quotient of the complex ball. By applying our method directly to Poincaré type cusps, we also get a partial localization result.