Kähler normal holomorphic coordinates ($h_{i\bar{j}}={\delta }_{ij}+O(|z{|}^2)$)

At any point of a Kähler manifold, a complex-linear coordinate change normalizes the Hermitian metric matrix. Closedness makes its holomorphic first derivatives symmetric in the two unbarred indices. A quadratic holomorphic coordinate change removes those derivatives, and Hermitian symmetry removes the antiholomorphic derivatives. Conversely, having this first-order normalization at every point makes the fundamental Hermitian form closed.