Moment maps, symplectomorphism groups and compatible complex structures

Abstract

In this paper, we apply Donaldson's general moment map frame-work for the action of a symplectomorphism group on the correspond-ing 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 struc-tures 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.