Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein

Question

Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?

I was told that we can use the following method:

Step 1: Use Bochner tecnique to prove $\dim H^2=1$

Step 2: Constant scalar curvature $\implies$ Ricci curvature is coclosed.

Step 3: Using Hodge theorem to derive the result.

Related reference:

On the Second Cohomology Group of a Kaehler Manifold of Positive Curvature

How can we prove this in detail? Or where can I find further reference for this?

Update:

I heard this is a famous result from Hongxi Wu and his partner, and it can be proved using Weitzenbock method.

Answer 1

I assume your manifold is compact.

It follows from Siu-Yau's proof of the Frankel conjecture that your manifold $X$ is biholomorphic to projective space (see also Mori's work) [1]. In particular, $\dim H^2(X,\mathbb{R})=1$, and so $c[\omega] = \mathcal{O}(1)$ for some $c \in \mathbb{R}_{>0}$. A Kähler metric of constant scalar curvature satisfying these cohomological conditions must be Kähler-Einstein, by the steps (2) and (3) you suggest [p60, 2]. In particular, $\omega$ must be the Fubini-Study metric by uniqueness of Kähler-Einstein metrics [3].

The difficult part here is really Siu-Yau. I am not sure if there is an easier way to obtain your desired conclusion. The Siu-Yau result is only used to get $\dim H^2(X,\mathbb{R})=1$, once you have this the result follows fairly directly.

[1] Siu-Yau. Compact kähler manifolds of positive bisectional curvature.

[2] Székelyhidi. An introduction to extremal Kähler metrics.

[3] Bando-Mabuchi. Uniqueness of Einstein Kähler metrics modulo connected group actions.