Compatible complex structures on symplectic rational ruled surfaces

Abstract: In this paper 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 $\cat J_\omega$ 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.