Szegő Kernels and Poincaré Series

Let M = M̃/Γ be a Kähler manifold, where Γ ' π1(M) and where M̃ is the universal Kähler cover. Let (L, h) → M be a positive Hermitian holomorophic line bundle. We first prove that the L Szegő projector Π̃N for L holomorphic sections on the lifted bundle L̃ → M̃ is related to the Szegő projector for H(M,L ) by Π̂N (x, y) = ∑ γ∈Γ ̃̂ ΠN (γ · x, y). We apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of M̃ with respect to L̃ and to surjectivity of Poincaré series. Let (M,ω) denote a compact Kähler manifold of dimension m, and let (M̃, ω̃) denote its universal Kähler cover with deck transformation group Γ = π1(M). We assume that Γ is an infinite group so that M̃ is complete noncompact. Let (L, h) → (M,ω) denote a positive hermitian line bundle and let (L̃, h̃) be the induced hermitian line bundle over M̃ . The first purpose of this note is to prove that for sufficiently large N ≥ N0(M,L, h), the Szegő kernel of the holomorphic projection ΠhN : L 2(M,LN )→ H0(M,LN ) on the quotient is given by the Poincaré series of the Szegő projection for L2 holomorphic sections on the universal cover (Theorem 1). This relation is standard in the theory of the Selberg trace formula on locally symmetric spaces, but seems not to have been proved before in the general setting of positive line bundles over Kähler manifolds. As will be seen, it is a consequence of standard Agmon estimates on off-diagonal decay of the Szegő kernel [Del, L] and of the local structure of the kernel given by the Boutet de Monvel-Sjöstrand parametrix for both Szegő kernels [BouSj, BBSj]. This relation is then used to simplify and unify a number of results on universal covers of compact Kähler manifold. One application is a short proof of the holomorphic convexity with respect to the positive line bundle (L̃, h̃) (Theorem 2) proved by T. Napier [N]. A second application is a simple proof of surjectivity of Poincaré series (Theorem 3). The problem of determining the least N0(M,L, h) for which these results are true is not treated in this article. To state the results, we need to introduce some notations. For any positive hermitian line bundle (L, h)→ (M,ω) over a Kähler manifold, we denote by H0(M,LN ) the space of holomorphic sections of the N -th power of L. We assume throughout that ω := − i π∂∂̄ log h is a Kähler metric. The Hermitian metric h induces the inner Date: August 12, 2013.