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Summary: MOMENT MAPS, SYMPLECTOMORPHISM GROUPS AND
COMPATIBLE COMPLEX STRUCTURES
MIGUEL ABREU, GUSTAVO GRANJA, AND NITU KITCHLOO
Abstract. In this paper we apply Donaldson's general moment map frame-
work for the action of a symplectomorphism group on the corresponding space
of compatible (almost) complex structures to the case of rational ruled sur-
faces. This gives a new approach to understanding the topology of their sym-
plectomorphism 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 symplec-
tomorphism group, the deformation theory of compatible complex structures
and the groups of complex automorphisms of these complex structures.
1. Introduction
The known results regarding the topology of symplectomorphism groups, for
2-dimensional surfaces and 4-dimensional rational ruled surfaces, indicate a direct
relation with the topology of the corresponding groups of complex automorphisms.
The goal of this paper is to formulate this empirical relation more precisely, within
a general framework involving infinite dimensional groups, manifolds and moment
maps. This general framework goes back to Atiyah and Bott [AB], was made
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