Moment maps, symplectomorphism groups and compatible complex structures (Q2642406)

In the paper under review one describes two applications of the infinite-dimensional moment map framework developed by \textit{S.~Donaldson} [in: M. F.~Atyiah, D.~Iagolnitzer (eds.), Fields Medallists Lectures (World Sci., Singapore) 384--403 (1997; Zbl 0913.01020)] for the action of a symplectomorphism group on the corresponding space of compatible almost complex structures. To state the main theorems of the present paper, let us denote by \(\sigma\) the standard area form on the sphere \(S^2\). For \(\lambda\in{\mathbb R}\) with \(\lambda\geq1\) let \(\omega_\lambda=\lambda\sigma\oplus\omega\), which is a symplectic form on \(S^2\times S^2\), and denote by \(G_\lambda\) the symplectomorphism group of \((S^2\times S^2,\omega_\lambda)\). With this notation, the first of the theorems announced in the paper under review says that the space \(X_\lambda\) of compatible (integrable) complex structures on \((S^2\times S^2,\omega_\lambda)\) is weakly contractible, which implies that \(H^*_{G_\lambda}(X_\lambda;{\mathbb Z})\simeq H^*(\text{B}G_\lambda;{\mathbb Z})\). According to the second theorem announced here, for every natural number \(\ell\) and every \(\lambda\in(\ell,\ell+1]\) there exists a group isomorphism \[ H^*(\text{B}G_\lambda;{\mathbb Z})\simeq H^*(\text{BSO}(3)\times\text{BSO}(3);{\mathbb Z}) \oplus\bigoplus\limits_{k=1}^\ell\Sigma^{4k-2} H^*(\text{B}S^1\times\text{BSO}(3);{\mathbb Z}). \] The details of the proofs for these two theorems are postponed for a forthcoming paper. As regards the present exposition, it should be noted that it is very clear and is illustrated by several interesting examples.