A remark on the gradient map

Abstract

For a Hamiltonian action of a compact group $U$ of isometries on a compact Kähler manifold $Z$ and a compatible subgroup $G$ of $U^{\mathbb{C}}$, we prove that for any closed $G$-invariant subset $Y\subset Z$ the image of the gradient map ${\mu }_{\mathfrak{p}}(Y)$ is independent of the choice of the invariant Kähler form $\omega$ in its cohomology class $[\omega ]$.