Explicit local form for Kahler metrics In Huybrechts "Complex Geometry" exercise 1.3.5. says show that $\frac{i}{2\pi}\partial\bar{\partial}\text{log}(|z|^2+1) \in \mathcal{A}^{(1,1)}(\mathbb{C})$ is the fundamental form of a compatible metric $g$ that oscillates to order two at any point.  
First, I have no idea where the logarithm is coming from and its driving me crazy. Does it oscillating to order two at any point give a differential equation who only solution is such a log? Also, it appears to me that $\frac{i}{2\pi}\partial\bar{\partial}\text{log}(|z|^2+1)$ is $0$ at the origin and therefore does not look like $1+O(|z|^2)$ there.
 A: Assuming I can differentiate correctly, the Kahler form is
\begin{align*} \omega & =  \frac{i}{2\pi} \partial\bar{\partial} \log ( |z|^2 + 1 ) 
\\ & = \frac{i}{2\pi} \partial \left( \frac{z}{|z|^2 + 1}d\bar z\right)
\\ & = \frac{i}{2\pi} \left( \frac{1}{|z|^2 + 1} dz \wedge d\bar{z} -  \frac{|z|^2}{(|z|^2 + 1)^2} dz \wedge d\bar{z} \right).
\\ & = \frac{i}{2\pi} \frac{1}{(|z|^2 + 1)^2} dz \wedge d\bar{z}.\end{align*}
So the metric is
$$ ds^2 =  \frac{1}{2\pi} \frac{1}{(|z|^2 + 1)^2} dz \otimes d\bar{z},$$
(up to factors of $i$ and $2$ that I haven't been careful with).
Anyway, you can expand this around the origin as
$$ ds^2 = \frac{1}{2\pi} \left( 1 - 2|z|^2 + \dots \right) dz \otimes d\bar{z},$$
and you can see that it differs from the flat metric $ds^2 = \frac{1}{2\pi}  dz \otimes d\bar{z}$ only at order two in $z$ and $\bar z$.
[By the way, if you add a point at infinity, your manifold $\mathbb{C}$ turns into a $\mathbb{CP}^1$, which is the same as a two-sphere via stereographical projection. The metric we computed is the round metric on the two-sphere.]
Edit: To address the question of whether this is the unique metric that osculates to order 2 everywhere, the answer is no. The flat metric,
$$ ds^2 = dz \otimes d \bar z,$$
clearly osculates to order 2 everywhere, but its Kahler form is $\omega = \partial \bar{\partial} |z|^2$.
Moreoever, the condition of the metric osculating to order two is equivalent to the condition that the Kahler form is closed. But on a one-dimensional complex manifold, the Kahler form $\omega$ is guaranteed to be closed, since $d\omega$ is a three-form, and there are no non-trivial three-forms on one-dimensional complex manifolds! So every hermitian metric on a one-dimensional complex manifold osculates to order two.
