Three problems on the group of symplectomorphisms (Q1200382)

Let \((X,\omega^ 2)\) be a symplectic manifold, \(\dim X = 2n < \infty\), and let \({\mathcal D}_ 0 = {\mathcal D}_ 0(X,\omega^ 2)\) be the identity component of the group of all diffeomorphisms \(x \mapsto x \cdot g\), \(x \in X\), preserving \(\omega^ 2\) and having (in the case of noncompact \(X\)) compact support. Thus, there arises the Calabi homomorphism \(\alpha = \alpha_ X: {\mathcal D}_ 0 \to H^ 1_ 0(X,\mathbb{R})\), where \(H^ 1_ 0(X,\mathbb{R})\) denotes the cohomology space with compact supports [see also \textit{E. Calabi}, in Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 1-26 (1970; Zbl 0209.258)]. Let \({\mathcal D}^ 0_ 0 = \text{Ker }\alpha\). The author demonstrates three theorems connected with the groups \({\mathcal D}_ 0\) and \({\mathcal D}^ 0_ 0\), the symplectic manifolds considered being finite-dimensional (first and third problems) or infinite-dimensional (second problem).