Szego kernels and Poincare series

TL;DR

This paper establishes that the Szego kernel on a Kahler manifold can be expressed as a sum over the deck group of the universal cover, extending known results to more general settings and applying this to holomorphic convexity and Poincaré series.

Contribution

It proves the sum over Gamma formula for Szego kernels on general Kahler manifolds, filling a gap in the literature and simplifying related proofs.

Findings

Szego kernel quotient equals the periodization of the universal cover's Szego kernel.

Application to Napier's theorem on holomorphic convexity.

Simplification of Poincaré series discussion in Kollar's work.

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.