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.
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The first author is partially supported by NSF grants nos. DMS-12-06748 and DMS-1541126.
The second author is partially supported by NSF grant no. DMS-1510232.
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Lu, Z., Zelditch, S. Szegő kernels and Poincaré series. JAMA 130, 167–184 (2016). https://doi.org/10.1007/s11854-016-0033-9
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DOI: https://doi.org/10.1007/s11854-016-0033-9