Every holomorphic map between Kähler manifolds is harmonic I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section:

Every holomorphic map between Kähler manifolds is harmonic.

I am not that familiar with harmonic maps, so I haven't been able to figure out why this is true. A quick google search didn't provide me the answer either. Is there a simple explanation for why this is true? If not, does anyone know of a reference for this fact?
 A: If you are still looking, here is a nice reference:
Calculus of variations and harmonic maps-By Hajime Urakawa, see chapter 4, proposition 3.14 (page 146). It is a very interesting book in general.
A: An excellent reference for Kahler manifolds, maps, and specifically harmonic maps is James Eells and Luc Lemaire, "Two Reports on Harmonic Maps," which incidentally also contains a third report, published by World Scientific.
A: For the sake of completeness, I try to add a proof here
Let $F$, be a holomorphic map between compact Kahler manifolds $(M,J)$ and $(N,J')$, then $$\nabla F_*(X,JY)=\nabla_XF_*(JY)-F_*\nabla_X(JY)$$ but $$\nabla F_*(X,JY)=J'(\nabla F_*)(X,Y)$$
and $$\nabla F_*(JX,JY)=-\nabla F_*(X,Y)$$
Choose a local orthonormal frame $\{e_1,...,e_m,\bar{e_1},...,\bar{e_m}\}$ for $M$, such that $Je_i=\bar e_i$ for all $i$. Hence we obtain 
$$\tau(F)=\sum_{i=1}^m(\nabla F_*)(e_i,e_i)+(\nabla F_*)(\bar e_i,\bar e_i)=0$$
The map $F$ is said to be harmonic if its tension field
vanishes i.e. $\tau (F) = 0$.
See Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their ...
By Bayram Sahin
