= Solution
Conversely, choose the stated normal holomorphic coordinates at an arbitrary point. The coefficient matrix equals $I+O(|z|^2)$, so all of its first derivatives at the center vanish. Since the coordinate differentials themselves are closed,
$$
d\omega(0)=\frac i2\sum_{i,j}dh_{i\bar j}(0)\wedge dz_i\wedge d\bar z_j=0.
$$
Every point can serve as the center, hence \b[$d\omega=0$ on all of $X$]. Combined with the preceding construction, this proves the equivalence of (a) and (b); it is a first-jet characterization rather than a claim that the metric is flat on a neighborhood.
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