On the geometry of moduli spaces of symplectic structures
Let \({\mathcal S}(M)\) be the space of all smooth symplectic structures on a manifold \(M\). The diffeomorphism group \({\mathcal D}(M)\) and its identity component \({\mathcal D}_0(M)\) act on \({\mathcal S}(M)\) and provide the moduli spaces \({\mathcal M}(M)={\mathcal S}(M)/{\mathcal D}(M)\), \({\mathcal M}_0(M)={\mathcal S}(M)/{\mathcal D}_0 (M)\). First, the authors prove that the space \({\mathcal S}_{\text{lin}}(V)\) of all symplectic forms on a real vector space \(V\) of dimension \(2n\) is a homogeneous pseudo-Riemannian manifold of signature \((n^2,n^2-n)\). Secondly, the \(L^2\)-version of the metric in the linear case yields a symmetric tensor field \(\widetilde g\) on \({\mathcal S}(M)\) and, by applying symplectic Hodge theory, a symmetric tensor field on \({\mathcal M}_0(M)\). If a certain \(\omega\in{\mathcal S}(M)\) admits a compatible complex structure \(J\) with \((\omega,J)\) a Kähler structure, then \(g_{[\omega]}\) is nondegenerate. Third, the intrinsical metric \(g\) for the manifold \(M=(S^2)^m\) is determined. All fundamental concepts are clearly explained, the article is self-contained.
scientific article; zbMATH DE number 1889167