 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,Every point can serve as the center, hence $d\omega =0$ 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.