Symplectic and Kähler structures on $\mathbb{C}{P}^1$-bundles over $\mathbb{C}{P}^2$

Abstract

We show that there exist symplectic structures on a $\mathbb{C}{P}^1$-bundle over $\mathbb{C}{P}^2$ 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 $\mathbb{C}{P}^1$-bundle over $\mathbb{C}{P}^2$ by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $\mathbb{C}{P}^2$.

Accepted by Selecta

Cited by in corpus (1)

• Hamiltonian $S^1$-actions on complete intersections