Compatible complex structures on symplectic rational ruled surfaces

Abstract

In this article, we study the topology of the space Iω of complex structures compatible with a fixed symplectic form ω, using the framework of Donaldson. By comparing our analysis of the space Iω with results of McDuff on the space Jω of compatible almost complex structures on rational ruled surfaces, we find that Iω 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