Bergman kernels and Poincar\'e series

Abstract

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\widetilde{X}$ with bounded geometry by a discrete group $\Gamma$ of its isometries is the same as the averaging over $\Gamma$ of the Bergman kernel on $\widetilde{X}$. We then use these results when $\widetilde{X}$ is a Hermitian symmetric space to show that a large class of relative Poincar\'e series does not vanish. This extends the results of Borthwick-Paul-Uribe and Barron (formerly Foth) to the case of general locally symmetric spaces of finite volume.