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arXiv · 2603.04842

Bergman kernels and Poincar\'e series

Louis Ioos·Wen Lu·Xiaonan Ma·George Marinescu
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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.

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Louis Ioos, Wen Lu, Xiaonan Ma, George Marinescu. 2026-03-05. Bergman kernels and Poincar\'e series. https://arxiv.org/abs/2603.04842

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