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 (Gomf) 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?

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?