I guess you are right. If the metric is Kähler they coincide, otherwise not. SP let's suppose the metric is Kähler or that we are talking about holomorphic scalar curvature...

That's exactly what I do.

Sorry, what's the holomorphic scalar curvature? Did you mean holomorphic sectional curvature? If yes, I don't think you are right: they are simply different objects... The first one is a tensor-like object the second one a function.

Anyway, I just need it in the Kähler case.

Actually, the answer is not in the pages you indicated. It is given as an exercise at the end of that chapter. I'll try to work it out and possibly post it here later.

Hey little brother, thank you very much! I go and check !