In this article, we study the topology of the space of complex structures compatible with a fixed symplectic form , using the framework of Donaldson. By comparing our analysis of the space with results of McDuff on the space of compatible almost complex structures on rational ruled surfaces, we find that 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
"Compatible complex structures on symplectic rational ruled surfaces." Duke Math. J. 148 (3) 539 - 600, 15 June 2009. https://doi.org/10.1215/00127094-2009-033