Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

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