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?



Independence of Duistermaat-Heckman measure
We just need to show that $(X,\omega)$ and $(X,\omega\_\phi)$ are $T$-equivariant symplectomorphic, then there is a commutative diagram
relating the moment maps $J$ and $J\_\phi$ over the same polytope through the equivariant symplectomorphism, and this directly implies their induced Duistermaat-Heckman measure agree. 


We apply Moser's argument. Let $\omega\_s=(1-s)\omega+s\omega\_\phi$. Then $\omega\_s-\omega=sd(\Re i\overline{\partial}\phi)$. Let $Y\_s$ be the vector field satisfying $\iota\_{Y\_s}\omega\_s=-\Re i\overline{\partial}\phi$, and let $\varphi\_Y^s$ be the time $s$ flow for $Y\_s$. Then we have $$\frac{d}{ds}(\varphi\_Y^s)^\*\omega\_s=(\varphi\_Y^s)^\*(\mathcal{L}\_{Y\_S}\omega\_s+\frac{d}{ds}\omega\_s)=(\varphi\_Y^s)^\*(d\iota\_{Y\_s}\omega\_s+d\Re i\overline{\partial}\phi)=0.$$
Hence $(\varphi\_Y^1)^\*\omega\_\phi=\omega$. Since $\varphi\_Y^1$ commutes with the $T$-action (because by definition $Y\_s$ is $T$-invariant), it gives a $T$-equivariant symplectomorphism between $(X,\omega)$ and $(X,\omega\_\phi)$. 


