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?