Moser type theorem for toric hyper-Kähler quotients (Q1592621)

\textit{J. Moser} proved in [Trans. Am. Math. Soc. 120, 286--294 (1965; Zbl 0141.19407)] that the elements of a smooth family \(\{\omega_t\}_{0\leq{}t\leq1}\) of mutually cohomologous symplectic forms on a closed manifold \(M\) belong to the same diffeomorphism class, i.\,e., \(\Phi_t^\ast\omega_t=\omega_0\) for some diffeomorphism \(\Phi_t\) of \(M\). For the prove of this result the assumption of compactness of \(M\) assures the completeness of certain vector fields \(Z_t\) constructed in order to establish the family of diffeomorphisms \(\Phi_t\) of \(M\). Here, an analog of this theorem is proved for not necessarily compact hyper-Kähler quotients under a certain additional assumption. In more detail: Let \((M,g,I,J,K)\) be a hyper-Kähler manifold, where \(g\) is a complete Riemannian metric, \(I,J,K\) parallel almost complex structures on \(M\) satisfying the ordinary rule of multiplication of quaternions and so that \(g\) is Hermitian with respect to \(I,J,K\). They induce three Kählerian 2-forms \(\omega_I,\omega_J,\omega_K\). For the action of a compact connected Lie group \(G\) with Lie algebra \({\mathfrak g}\) consider a moment map \[ \mu=\left(\mu_I,\mu_J,\mu_K,\right):M\to {\mathfrak g}^\ast\times{\mathfrak g}^\ast\times{\mathfrak g}^\ast \] and the inverse image \(M^\xi=\mu^{-1}(\xi)\subset M\) of a regular, \(G\)-invariant value \(\xi=\left(\xi_I,\xi_J,\xi_K\right)\) of \(\mu\). The quotient \(M^\xi/G\) is called a hyper-Kähler quotient of \(M\) and \(\omega_I,\omega_J,\omega_K\) induce symplectic forms on \(M^\xi/G\) denoted by the same symbols. The main result states that the diffeomorphism class of the symplectic manifold \(\left(M^\xi/G,\omega_I\right)\) does not depend on \(\left(\xi_J,\xi_K\right)\), while it may depend on \(\xi_I\). For the proof, again, completeness of a certain vector field must be assured. Considering linear actions of toric groups on quaternionic vector spaces non-compact examples are given where these assumptions are satisfied.