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?
Best Answer
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.
See page 107 of Griffiths--Harris, Principles of Algebraic Geometry.
Also, integrable complex structure is equivalent to the existence of local holomorphic coordinates by the Newlander-Nirenberg theorem.