  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 $\mathbb CP^{2}$.
