Some remarks about 1-convex manifolds on which all holomorphic line bundles are trivial

We give an abridged proof of an example already considered in [M. Colţoiu, On 1-convex manifolds with 1-dimensional exceptional set, Rev. Roumaine Math. Pures et Appl. 43 (1998) 97–104] of a 1-convex manifold X of dimension 3 such that all holomorphic line bundles on X are trivial. We also point out several mistakes of [Vo Van Tan, On the quasiprojectivity of compactifiable strongly pseudoconvex manifolds, Bull. Sci. Math. 129 (2005) 501–522] concerning this topic.