In this talk I will explain how a general moment map formal set-up, developed by Donaldson for the action of a symplectomorphism group on the space of compatible complex structures, can be used to interpret and extend known results on the topology of symplectomorphism groups of rational ruled surfaces. The actual proofs are based on a result of independent interest: the space of integrable complex structures compatible with a fixed (but arbitrary) symplectic form on a rational ruled surface is contractible. This is joint work with Gustavo Granja and Nitu Kitchloo.