Normal coordinates in Kähler manifolds and the background field method

Riemann normal coordinates (RNC) are unsuitable for Kähler manifolds since they are not holomorphic. Instead, Kähler normal coordinates (KNC) can be defined as holomorphic coordinates. We prove that KNC transform as a holomorphic tangent vector under holomorphic coordinate transformations, and therefore that they are natural extensions of RNC to the case of Kähler manifolds. The KNC expansion provides a manifestly covariant background field method preserving the complex structure in supersymmetric nonlinear sigma models.