Moment maps and Jacobian modules (Q1973269)

Let \((M,\omega)\) be a symplectic manifold with an action of a compact Lie group \(G\) which preserves the symplectic form. When the action is Hamiltonian there is a moment map \(\phi: M\to{\mathfrak g}^*\) where \({\mathfrak g}^*\) is the dual of the Lie algebra \({\mathfrak g}\) of \(G\). In [J. Differ. Geom. 49, No. 1, 183-201 (1998; Zbl 0920.57010)], \textit{Y. Karshon} has given a definition of an abstract moment map without reference to symplectic structures. The present paper is devoted to the question whether an abstract moment map can locally be written as a Hamiltonian one. This question has, for torus actions, been answered affirmative by V. Ginzburg, V. Guillemin, and Y. Karshon in a recent preprint. In the present paper, the author considers the case \(G=\text{SU}(2)\), the simplest non-abelian case. The author considers a linear action of \(G\) on some finite-dimensional vector space \(V\). Then, a polynomial moment map \(\phi\) is seen to be Hamiltonian iff the image of the complexification \(\phi_{\mathbb{C}}\) is contained in the image of the dual complexified Jacobian map of the action.