King's College London

We show that there exist symplectic structures on a CP1-bundle over CP2 2 that do
3 not admit a compatible Kähler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian T2 4 -symmetry. Tolman’s manifold was shown to be diffeomorphic to a CP1-bundle over CP2 5 by Goertsches, Konstantis, 6 and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over CP2