1-convex and holomorphically convex

Question

A complex manifold $M$ is called $1$-convex if there exists a smooth, exhaustive, plurisubharmonic function that is strictly plurisubharmonic outside a compact set of $M$.

Can we prove that if $M$ is $1$-convex, then $M$ is holomorphically convex, i.e., the holomorphic hull of any compact set is compact?

Answer 1

The answer to your question is yes , 1-convex implies holomorphic convexity. This is Grauert's solution of the Levi problem.You can find a proof in the book of Fritzsche and Grauert titled From Holomorphic functions to Complex manifolds pages 289 to 294 .