Symplectic and K\xe4hler structures on mathbb CP^1-bundles over mathbb CP^2

Abstract

We show that there exist symplectic structures on a mathbb {CP}^1-bundle over mathbb {CP}^2 that do not admit a compatible Kähler 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}.