Kählerian reduction

We will consider a Hamiltonian action of a non compact group $G$ consisting of holomorphic Kähler isometries on a Kählerian manifold $X$. If the zero fiber $\mathcal M$ of the corresponding momentum map is non empty, then the quotient $\mathcal M/G$ is know to be a Kähler space. We will show that locally near $\mathcal M$ the Kählerian form $\omega$ on $X$ has a $G$-invariant Kähler potential, i.e. it is given in a $G$-invariant neighborhood by an invariant strictly plurisubharmonic function $\rho$ which determines the momentum map.