In this paper, we apply Donaldson's general moment map framework for the action of a
symplectomorphism group on the corresponding space of compatible (almost) complex structures
to the case of rational ruled surfaces. This gives a new approach to understanding the
topology of their symplectomorphism groups, based on a result of independent interest: the
space of compatible integrable complex structures on any symplectic rational ruled surface is
(weakly) contractible. We also explain how in general, under this condition, there is a direct
relationship between the topology of a symplectomorphism group, the deformation theory of
compatible complex structures and the groups of complex automorphisms of these complex
structures.