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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.

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    $\begingroup$ A very simple example is $\mathbb{CP}^1 \times \mathbb{CP}^1$, but in general one should expect a very broad class (though not all) of Kahler manifolds to admit Einstein metrics. On the other hand, a metric has constant holomorphic sectional curvature only if it is covered by $\mathbb{CP}^n$, $\mathbb{C}^n$ or $\mathbb{CH}^n$ (I.e., the complex space forms). $\endgroup$
    – Gabe K
    Commented Aug 13 at 12:02
  • $\begingroup$ @GabeK I guess the reference is Griffith& Harris, yes? $\endgroup$
    – Ali Taghavi
    Commented Aug 13 at 12:24
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    $\begingroup$ It should definitely be there, but I don’t have a copy on hand to verify. The intuition that a broad class of metrics admit KE metrics is that it should be equivalent to some algebraic condition and I’ve heard experts say that “most” manifolds should satisfy it. However, this is not my area of expertise and there are certainly counterexamples, so I’m not sure how precise this is. $\endgroup$
    – Gabe K
    Commented Aug 13 at 12:51
  • $\begingroup$ @GabeK Yes I see, thank you $\endgroup$
    – Ali Taghavi
    Commented Aug 13 at 12:52

1 Answer 1

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Answer to expand on my comment:

A very simple example is $\mathbb{CP}^1 \times \mathbb{CP}^1$, but in general one should expect a very broad class (though not all) of Kahler manifolds to admit Einstein metrics. On the other hand, a metric has constant holomorphic sectional curvature only if it is covered by $\mathbb{CP}^n$, $\mathbb{C}^n$ or $\mathbb{CH}^n$ (I.e., the complex space forms).

To give a non-trivial example with positive curvature, take your favorite Fano complex surface (a del Pezzo surface). There are 10 deformation families of these and Tian showed that a smooth Fano surface admits a Kähler–Einstein metric if and only if it is not the blow up of the complex projective plane in one or two points.

More generally, there is a deep result of Chen-Donaldson-Sun that a Fano variety admits a KE metric whenever it satisfies an algebraic condition called $K$-polystability, so to find examples for your question it suffices to find Fano manifolds which satisfy this condition but aren’t projective space.

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  • $\begingroup$ Thank you for your answer. I need time to understand it. BTW by cover you mean universal cover? What reference for the first paragraph? $\endgroup$
    – Ali Taghavi
    Commented Aug 13 at 12:16
  • $\begingroup$ I know (without proof) the similar real case the covering by spher, plane and hyperbolic space. $\endgroup$
    – Ali Taghavi
    Commented Aug 13 at 12:29
  • $\begingroup$ i know that $S^2\times S^2$ is covering of itself so impossible to be covered by the three space you mentioned so I think I got my complete answer thanks for that. The only remaining point: Was the reference Griffith &Harris? $\endgroup$
    – Ali Taghavi
    Commented Aug 13 at 12:44

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