# Riemannian metric of a Kähler manifold

In p.52 of Morgan's book on Seiberg-Witten equations, there is the following paragraph:

$$(\cdots)$$ Assume that $$X$$ is a complex manifold with a Kahler metric. This means that $$X$$ has a Riemannian metric in which the $$J$$-operator of the complex structure is orthogonal, but also that at each point of $$X$$ there is a local holomorphic coordinate system in which the Riemannian metric is standard to second order. $$(\cdots)$$

What does "standard to second-order" means? Also, how does the usual definition of Kahler manifolds (symplectic manifold with a compatible, integrable almost complex structure) imply the existence of such local holomorphic coordinate system?

• One can also control parts of the higher derivatives of the metric: see here Commented Jul 27, 2021 at 13:39

In normal coordinates, a Kähler metric $$g$$ satisfies $$g_{i \overline{j}}(p)= \delta_{ij}$$ and $$\partial_k g_{i \overline{j}}(p)=0$$. In particular, a Kähler metric agrees with the standard Euclidean metric up to second order.