  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.
