let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as $$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{a}$$ where $\mathfrak{h}$ is the space of holomorphic vector fields vanishing somewhere on $M$. Suppose $X\in \mathfrak{h}$ has a Hamiltonian potential $\varphi_{X}\in C^{\infty}(M,\mathbb{R})$ $$\overline{\partial}\varphi_{X}=-\frac{1}{2}i_{X}\omega$$ Now we blow up $M$ at some points $p_{1},\ldots,p_{n}$ and we denote with $\tilde{M}$ this blow up and with $\pi$ the canonical surjection on $M$. Suppose $X$ lifts to $\tilde{M}$ i.e. $X$ vanishes at $p_{1},\ldots,p_{n}$. Now we pick a Kähler metric $\omega_{\varepsilon}\in[\omega_{\varepsilon}]$

$$[\omega_{\varepsilon}]:=\pi^{*}[\omega]+\varepsilon\sum_{i=1}^{n}c_{1}(\mathcal{O}(-E_{i}))$$ with $E_{i}$ the exceptional divisors at points. We DO NOT assume that $\omega_{\varepsilon}$ is invariant for the flow of the lift of $X$.

The question is the following: is there always a Hamiltonian potential $\tilde{\varphi}_{X}\in C^{\infty}(M,\mathbb{R})$ such that

$$\overline{\partial}\tilde{\varphi}_{X}=-\frac{1}{2}i_{X}\omega_{\varepsilon}$$
or there is some obstruction?