Moment maps, symplectomorphism groups and compatible complex structures

Abstract

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.

A personal tour through symplectic topology and geometry

“…One needs to check that standard deformation theory can in fact be used here, in the context of compatible complex structures. This, together with points (vi) and (viii), is proved in [7]. Point (iv) is proved in [12].…”

“…It implies, by standard equivariant cohomology theory (see [7]), the following corollary. Contractibility of X λ .…”

“…(ii) Atiyah-Guillemin-Sternberg's Convexity Theorem for the moment map of Hamiltonian torus actions ( [9,25]) and Kähler geometry of toric orbifolds in symplectic coordinates (sections 4 and 5, references [3,4,5]). (iii) Donaldson's moment map framework for the action of the symplectomorphism group on the space of compatible almost complex structures ( [17]) and the topology of the space of compatible integrable complex structures of a rational ruled surface (sections 6 and 7, references [6,7]). …”

Loops in the Hamiltonian group: a survey

“…(This was first proved by Audin [4] and Ahara-Hattori [3]; see also Karshon [19].) Moreover the homotopy type of Ham is understood when M = CP 2 or S 2 × S 2 or a one point blow up of such; see for example Gromov [15], Abreu-McDuff [2], Abreu-GranjaKitchloo [1] and Lalonde-Pinsonnault [28]. For work on nonHamiltonian S 1 actions in dimensions 4 and above see Bouyakoub [6], Duistermaat-Pelayo [7] and Pelayo [50].…”

Compatible complex structures on symplectic rational ruled surfaces

“…He also showed that, restricted to the space I ω of compatible integrable structures, this action fits into the general framework of infinite dimensional geometric invariant theory, going back to Atiyah and Bott [AB1] (see [AK1 ] for more discussion of this point of view). Therefore, in principle, the norm square of the moment map should induce a stratification of I ω with critical points being the extremal Kähler metrics compatible with the symplectic form.…”