Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is complete.
It is known that if $X$ is (biholomorphic to) a bounded domain in some $\mathbb C^n$, then it is holomorphically convex, and hence Stein.
In 1962, S. Kobayashi asked in this paper whether in general a Bergman complete complex manifold is necessarily holomorphically convex.
I was wondering what is the state of the art around this question.
My motivation comes in particular from the Shafarevich conjecture: the universal cover of a complex projective manifold should always be holomorphically convex.
Now, suppose you are given a complex projective manifold $X$ whose universal cover $\tilde X$ admits the Bergman metric $ds_{\tilde X}^2$. Then, by invariance under the deck transformation group, it descends to a Kähler metric on $X$ which is compact and hence $\tilde X$ is Bergman complete.
Thus, an affirmative answer to Kobayashi's question would confirm the Shafarevich conjecture at least in this (very) particular case.
