Normal Coordinates in Kahler Manifolds and the Background Field Method
  Riemann normal coordinates (RNC) are unsuitable for \kahler manifolds since they are not holomorphic. Instead, \kahler 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 \kahler manifolds. The KNC expansion provides a manifestly covariant background field method preserving the complex structure in supersymmetric nonlinear sigma models. 
Kiyoshi Higashijima, Etsuko Itou (Osaka Univ.) and Muneto Nitta   (Purdue Univ.)