= Pluriharmonic function
{title2=$\partial\bar\partial f=0$}
A smooth real function on a <complex manifold> is pluriharmonic if $\partial\bar\partial f=0$. A complex-valued function satisfies the same equation exactly when both real and imaginary parts are pluriharmonic. Restrictions to complex coordinate lines are <harmonic functions>. On a compact connected <complex manifold>, choose a maximum of $|f|$ and apply the harmonic maximum principle successively along coordinate lines; the function is constant on a polydisc, then on the manifold by open-and-closed propagation. Without connectedness, constancy is componentwise.
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