Suppose that a compact Kähler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth function from $PSH_{tor}(X,\omega) := \{\phi \in PSH(X,\omega)\;|\; \phi\text{ - invariant}\}$ the form $\omega_\phi := \omega + i\partial\bar{\partial}\phi$ is another invariant symplectic form in the same cohomology class, thus it has the same moment polytope.

Now the question is: if $J$ is the moment map for $\omega$ and $J_\phi$ is the moment map for $\omega_\phi$, do they produce the same Duistermaat-Heckman measure on the polytope? I suppose they do, but why?