On the coverings of proper families of 1-dimensional complex spaces.

Abstract

Proposition 1.3. Let π : X → T be a proper holomorphic surjective map of complex spaces, let t0 ∈ T be any point, and denote by Xt 0 := π−1(t0) the fiber of π at t0. Assume that dimXt 0 = 1. Let σ : X→ X be a covering space and let Xt 0 := σ−1(Xt 0). If Xt 0 is holomorphically convex, then there exist: (1) an open neighborhood D of t0; (2) a continuous plurisubharmonic vertical exhaustion function f : D := (π B σ)−1(D)→ R+ (i.e., the restriction of π B σ : D→ D to {f ≤ c} is proper for every c ∈R); and (3) an increasing sequence {aν}, aν → ∞, such that f is strongly plurisubharmonic near the level sets {f = aν}, ν ∈N. Remark1.4. This proposition is proved by Napier [5] for dimX = 2, dim T = 1, and X, T smooth.