My apology in advance if this question is obvious:

I know that an Einstein manifold need not have a constant sectional curvature example $\mathbb{C}P^n$. But this space has a constant holomorphic sectional curvature.

What Einstein manifold admit a holomorphic structure whose holomorphic sectional curvature is not constant?

Note: I admit that I do not know the answer to the above question. But after that I know an answer the next step would be the following: What is a manifold who admite Einstein structure and also holomorphic structure but does not admit simultaneously a Riemannian metric and a holomorphic structure for which the holomorphic sectional curvature would be constant. However I do not include this question to this post as original main question. Because I do not know even its elementary version, i.e, the main question.