# Kähler structure on open complex manifold

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?

• A Hopf surface with a point removed does not admit a Kähler metric, see this question. – Michael Albanese May 12 '17 at 11:10

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 mentioned by Michael Albanese in his comment, the standard complex structure on $M$ does not admit a compatible Kahler metric.
In this G&T paper there are examples of complex structures on $\Bbb R^4$ withouth compatible Kähler metrics. The point is that in this $(\Bbb R^4, J)$ there are holomorphic elliptic compact curves, so there are no compatible symplectic structures.