Szegő kernels and Poincaré series

Abstract

Let $M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }$ be a Kähler manifold, where Γ ~ π ₁ (M) and $\widetilde M$ is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L ² Szegő projector ${\widetilde \Pi _N}$ for L ²-holomorphic sections on the lifted bundle ${\widetilde L^N}$ is related to the Szegő projector for H ⁰(M, L ^N) by ${\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)$. We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of $\widetilde M$ with respect to ${\widetilde L^N}$ and to surjectivity of Poincaré series.