= Kähler normal holomorphic coordinates
{c}
{title2=$h_{i\bar j}=\delta_{ij}+O(|z|^2)$}
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.
Back to article page
