Thanks! The cohomology is almost the same as for the form. But the definition of moment map is always for the smooth symplectic form. I wonder whether it can be extended to the current.

Yes! But what I care more is the pointwise property of $X(f)$. And it seems impossible if we only know the Laplace of $f$.

Thank you for your reply! But I can not really understand the last sentence "(in particular you can do it if your vector field X has a zero, in which case X has a holomorphy potential)". Would you like to give some details? In my opinion, the difficulty of this problem is that we get less information from $g=\triangle f$, in particular, the Hessian of $f$.

I have added that $\gcd (m_1,m_2)=1$.

Sorry to provide the wrong link. Here is the right version.

projecteuclid.org/euclid.jdg/1214342145

In the problem, $\{(Y,y,\nu)\}$ is the convergence of $\{(M_i,m_i,Vol_i)\}$ in the sense of Gromov-Hausdorff distance, where $\nu$ and $Vol_i$ are volume forms with respect to $Y$ and $M_i$. $\Psi(\eta|\varepsilon,\psi,n)$ is a non-negative function which is $0$ while $\eta$ tends to $0$. In my opinion, the last equation is a contradiction to the result of the lemma3.1 i.e. if every minimal geodesic connecting $y_1$ and $y_2$ intersects $B$, then there exists positive number $c$ such that $c\leq \sum_j r_j^{-1}\nu(B_{r_j}(w_j))$.