A note on the moment map on compact K\"ahler manifolds

We consider compact K"ahler manifolds acted on by a connected compact Lie group $K$ of isometries in Hamiltonian fashion. We prove that the squared moment map $\mathrm{\parallel }\mu {\mathrm{\parallel }}^2$ is constant if and only if the manifold is biholomorphically and $K$-equivariantly isometric to a product of a flag manifold and a compact K"ahler manifold which is acted on trivially by $K$. The authors do not know whether the compactness of $M$ is essential in the main theorem; more generally it would be interesting to have a similar result for (compact) symplectic manifolds.