Extend Kähler form to a point

Question

I'm thinking about extending a Kähler form on manifold $M-x_0$ to $M$. Here $M$ is a complex closed manifold, and I equip $M-x_0$ a natural induced complex structure.

Following is my way of thinking:

I work on local holomorphic charts around $x_0$, Now consider function $f=\omega(X, JX)$ for any local holomorphic vector field $X$. If I can prove $\Delta f=0$, then,

1.If I can also control the speed that $f$ go to infinity, then I can remove the sigularity of $f$ at $x_0$.

2. by positivity of $f$ I can show $\omega$ is non degenerate.

But I don't know if $\Delta f=0$ is true or not, can anyone help me? If not, I wonder if there is any other approach to extend the Kähler form?

Thank you for your answer!