Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $L\_{X} g = 0$, where $L\_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian curvature tensor of $g$. Is $L\_{X} R = 0$? 



Lie derivative of curvature
$g\mapsto R(g)$ is a natural (non-linear) operation: it commutes with pullbacks by diffeomorphisms. 
This is just a transcription of: curvature transforms correctly under chart changes.
Thus we get 
$L\_X R(g) = dR(g)(L\_X g)$ which implies "Yes"; two earlier comments also said this. 


