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Moment maps, symplectomorphism groups and compatible complex structures

Research paper by Miguel Abreu, Gustavo Granja, Nitu Kitchloo

Indexed on: 19 Jul '05Published on: 19 Jul '05Published in: Mathematics - Symplectic Geometry



Abstract

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.