Holomorphic L2 Functions on Coverings of Pseudoconvex Manifolds

1. Let M be a complex manifold with a smooth boundary which will be denoted bM , dim C M = n. Let us denote M = M ∪ bM and assume for simplicity that M ⊂ ˜ M where˜M is a complex neighbourhood of M , dim C ˜ M = n, so that every point z ∈ bM is an interior point of˜M. Let us take a C ∞-function ρ : ˜ M → R such that M = z | ρ(z) < 0 , bM = z | ρ(z) = 0 ; dρ(z) = 0 for all z ∈ bM. (0.1) For any z ∈ bM denote by T c z (bM) the complex tangent space to bM : the maximal complex subspace in the real tangent space T z (bM), dim C T c z (bM) = n − 1. If z 1 ,. .. , z n are complex local coordinates iñ M near z ∈ bM , then T z ˜ M is identified with C n and T c z (bM) = w = (w 1 ,. .. , w n) n j=1 ∂ρ ∂z j (z)w j = 0. (0.2) The Levi form is an hermitian form on T c z (bM) defined in the local coordinates as follows: L z (w, ¯ w) = n j,k=1 ∂ 2 ρ ∂z j ∂ ¯ z k (z)w j ¯ w k. (0.3) The manifold M is called pseudoconvex if L z (w, ¯ w) ≥ 0 for all z ∈ bM and w ∈ T c z (bM). It is called strongly pseudoconvex if L z (w, ¯ w) > 0 for all z ∈ bM and all w = 0, w ∈ T c z (bM). In this case replacing ρ by e λρ − 1 with sufficiently large λ > 0 we can assume that L z (w, ¯ w) > 0 for all w = 0 (not only for w satisfying the condition in (0.2)). Equivalently, strongly pseudoconvex manifolds can be described as the ones which locally, in a neighbourhood of any boundary point, can be presented as strongly convex domains in C n. Denote by O(M) the set of all holomorphic functions on M .