Kähler metric ($g_{i\bar{j}}={\partial }_i{\partial }_{\bar{j}}K$)

Locally, a Kähler metric is the complex Hessian $g_{i\bar{j}}={\partial }_i{\partial }_{\bar{j}}K$ of a real Kähler potential. Adding a holomorphic function and its complex conjugate to $K$ leaves the metric unchanged.