A note on the moment map on compact Kähler manifolds

Details A note on the moment map on compact Kähler manifolds A note on the moment map on compact Kähler manifolds

2003-10-15

arxiv.org/abs/math/0310213v1

Mathematics

Symplectic Geometry

4 pages

Scientific paper

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 $\|\mu\|^2$ 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.

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