A note on the moment map on compact Kähler manifolds

الأصل · EN

We consider compact Kähler manifolds acted on by a connected compact Lie group K of isometries in Hamiltonian fashion. We prove that the squared moment map μ² is constant if and only if the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler 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.