Moment maps, symplectomorphism groups and compatible complex structures

Details Moment maps, symplectomorphism groups and compatible complex structures Moment maps, symplectomorphism groups and compatible complex structures

2005-07-19

arxiv.org/abs/math/0507382v1

Mathematics

Symplectic Geometry

To appear in the issue of the Journal of Symplectic Geometry devoted to the Stare Jablonki conference proceedings

Scientific paper

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.

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