Let $X$ a Kähler manifold and $f:X\to \mathbb C$ a function. $f$ is said to be pluriharmonic if its restriction to each curve in $X$ is harmonic.
Why is then $f$ harmonic? Does this simply follow from covering $X$ by curves? Does the same argument then imply that the restriction of $F$ to any complex submanifold of $X$ is harmonic? (I need to see that $f$ restricted to a special case of compact connected submanifolds is constant, which would follow from harmonicity)
On a similar note, I have seen another of definition of $f$ being pluriharmonic, namely $\partial \overline{\partial} f=0$ Then it follows from a simple calculation that $f$ is harmonic. But how do I see that this definition I equivalent?