Compatible complex structures on symplectic rational ruled surfaces

Abstract

In this article, we study the topology of the space $I_{\omega }$ of complex structures compatible with a fixed symplectic form $\omega$, using the framework of Donaldson. By comparing our analysis of the space $I_{\omega }$ with results of McDuff on the space $J_{\omega }$ of compatible almost complex structures on rational ruled surfaces, we find that $I_{\omega }$ is contractible in this case.

We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff