  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.
