  By revisiting Tian's peak section method, we obtain a localization principle
of the Bergman kernels on K\"ahler 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 K\"ahler-Einstein metric or when the manifold is a quotient of the
complex ball. By applying our method directly to Poincar\'e type cusps, we also
get a partial localization result.
