Symplectic and K\"ahler structures on $\mathbb CP^1$-bundles over $\mathbb CP^2$

TL;DR

This paper demonstrates the existence of symplectic structures on certain $ ext{CP}^1$-bundles over $ ext{CP}^2$ that cannot be endowed with compatible Kähler structures, using Mori theory and vector bundle facts.

Contribution

It provides the first explicit example of symplectic structures on these bundles lacking compatible Kähler structures, expanding understanding of symplectic versus Kähler geometry.

Findings

Existence of non-Kähler symplectic structures on $ ext{CP}^1$-bundles over $ ext{CP}^2$

Application of Mori theory to distinguish symplectic and Kähler structures

Identification of Tolman's symplectic structures as non-Kähler examples

Abstract

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 $\mathbb CP^{2}$.