Compatible complex structures on symplectic rational ruled surfaces

Abstract

In this paper we study the topology of the space $\I_ω$ of complex structures compatible with a fixed symplectic form $\omega$, using the framework of Donaldson. By comparing our analysis of the space $\I_ω$ with results of McDuff on the space $\cat J_ω$ of compatible almost complex structures on rational ruled surfaces, we find that $\I_ω$ 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.

Sign mistake in the formula for the cohomology in twisted case fixed. Reorganized sections 4 and 5 and added more detail to proofs. To appear in Duke Math. Journal