Szego kernels and Poincare series

Abstract

Let $M = \tilde{M}/\Gamma$ be a Kahler manifold, where $\tilde{M}$ is the universal Kahler cover, and where $\Gamma$ is the deck transformation group. Let $(L, h) \to M$ be a positive Hermitian holomorophic line bundle. Lift the Hermitian line bundle to $\tilde{M}$ and consider the relation between the orthogonal projection onto holomorphic sections on the quotient and onto $L^2$ holomorphic sections on $\tilde{M}$. We prove that the quotient Szego kernel is given by the periodization of the $L^2$ Szego kernel of the universal cover, i.e. its sum over the deck group $\Gamma$. Although this is a standard result for symmetric spaces (it is used in the Selberg trace formula) and is also standard for heat and wave kernels, it seems that a proof of the sum over Gamma formula was lacking for Szego kernels of positive line bundles over general Kahler manifolds. We apply the result to give a simple proof of Napier's theorem on the holomorphic convexity of $\tilde{M}$ with respect to $\tilde{L}^N$ and to surjectivity of Poincar\'e series. We also simplify the discussion of Poincare series in Kollar's book.