Szegő kernels and Poincaré series

Let $$M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }$$M=M˜/Γ be a Kähler manifold, where Γ ~ π1(M) and $$\widetilde M$$M˜ is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szegő projector $${\widetilde \Pi _N}$$Π˜N for L2-holomorphic sections on the lifted bundle $${\widetilde L^N}$$L˜N is related to the Szegő projector for H0(M, LN) by $${\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)$$Π^N(x,y)=∑γ∈ΓΠ^˜N(γ⋅x,y). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of $$\widetilde M$$M˜ with respect to $${\widetilde L^N}$$L˜N and to surjectivity of Poincaré series.