Is it always possible to put a compatible Kähler metric on an open complex manifold (non-compact, without boundary, say of finite topological type)?


**Edit 1**: Michael Albanese pointed out a nice counterexample (namely the punctured Hopf surface) when this is impossible for topological reasons. **Question**: Would it be possible to have counter-examples of the following unpleasant type: The underlying open manifold has a Kähler metric, but for a given complex structure we cannot find a compatible Kähler metric?


**Edit 2**: Misha pointed out there is a Kahler metric on punctured $S^3 \times S^1$. Thinking more about this, I cannot resist to ask one final variation. **Question** Does there exist an open complex manifold for which there is no Kahler metric on the underlying smooth manifold?



Kähler structure on open complex manifold
The punctured Hopf surface is also an example for the revised question. It is easy to see that the Hopf surface $X$ is parallelizable (since it is diffeomorphic to $S^3\times S^1$); hence, $M=X- \{x\}$ admits an immersion in $R^4={\mathbb C}^2$ (Hirsch-Smale theory). Now pull-back the standard (flat) Kahler structure from ${\mathbb C}^2$ to $M$ via the immersion. On the other hand, as noted in [this answer](https://mathoverflow.net/q/88996/21564) mentioned by Michael Albanese in his comment, the standard complex structure on $M$ does not admit a compatible Kahler metric. 


