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Let $M^{4}= \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$ and $\omega$ the symplectic form on $M^{4}$ given by the anticanonical polarisation.

Suppose we form a symplectic (Gompf) sum of two copies of $(M^{4},\omega)$ along a fibre of the $\mathbb{P}^{1}$-bundle (where the orientation reversing identification of the normal bundles is the obvious one i.e. we just pick an orientation reversing isomorphism of $\mathbb{R}^{2}$ and let the isomorphism of normal bundles be equal to this pointwise).

Question: Does the resulting symplectic $4$-manifold have a compatible Kahler structure?

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    $\begingroup$ I think if you Gompf sum along a fibre you just get the bundle over the connect sum of bases, i.e. the second Hirzebruch surface in this case. So yes, it has a compatible Kaehler structure. A more complicated example would be fibre summing elliptic surfaces along elliptic fibres: the result is still an elliptic surface (e.g. fibre sum of two rational elliptic surfaces is a K3). $\endgroup$
    – Jonny Evans
    Commented Feb 2, 2019 at 17:44
  • $\begingroup$ That makes sense, Thanks! $\endgroup$
    – Nick L
    Commented Feb 2, 2019 at 19:38

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