 Solution

The real coordinate functions are real and imaginary parts of the holomorphic coordinates $z_j$. Part (a), together with the real coefficients of the Laplace-Beltrami operator, therefore shows that every $x_l$ is a harmonic function. For the given divergence-of-gradient convention, applying the operator to $x_l$ givesThese are the harmonic coordinate identities. Expanding the operator on an arbitrary smooth function now givesThe bracketed coefficients vanish. Hence there is no first-derivative term in these coordinates:The reason is that the real components of holomorphic coordinates are harmonic coordinates for a Kähler metric; a general real coordinate change need not retain this simplification.