= 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 $x_l$ gives
$$
0=\Delta_dx_l=\frac1{\sqrt{\det g}}\sum_{k=1}^{2n}\frac{\partial}{\partial x_k}\left(\sqrt{\det g}\,g^{kl}\right).
$$
These are the <harmonic coordinate> identities. Expanding the operator on an arbitrary smooth function now gives
$$
\Delta_du=\sum_{k,l=1}^{2n}g^{kl}\frac{\partial^2u}{\partial x_k\partial x_l}
+\sum_{l=1}^{2n}\left[\frac1{\sqrt{\det g}}\sum_{k=1}^{2n}\frac{\partial}{\partial x_k}\left(\sqrt{\det g}\,g^{kl}\right)\right]\frac{\partial u}{\partial x_l}.
$$
The bracketed coefficients vanish. Hence \b[there is no first-derivative term in these coordinates]:
$$
\boxed{\Delta_du=\sum_{k,l=1}^{2n}g^{kl}\frac{\partial^2u}{\partial x_k\partial x_l}.}
$$
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.
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