Symplectic and K\"ahler structures on $\mathbb CP^1$-bundles over $\mathbb CP^2$

We show that there exist symplectic structures on a $\mathbb CP^1$-bundle over $\mathbb CP^2$ that do not admit a compatible K\"ahler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian $\mathbb T^2$-symmetry. Tolman's manifold was shown to be diffeomorphic to a $\mathbb CP^1$-bundle over $\mathbb CP^{2}$ by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $\mathbb CP^{2}$.