# Characterization of Kahler-Einstein manifolds

How can we characterize Kähler-Einstein manifolds by using cohomological method? More precisely How can we charactrize Fano Kähler-Einstein manifolds by using cohomological method? Any reference? I know that one of ways is by using spectrum.

There are some known results around 75, let me mention some of them.

Let $M$ be a compact Kähler manifold with a fundamental two form $\omega$, and Ricci 2-form $Ric(\omega)$, we say that $M$, is cohomologically Kähler-Einstein manifold if $[Ric(\omega)]=a[\omega]$ for some constant $a$. We write the Ricci tensor by $Ric=\sum_{\alpha,\beta} R_{\alpha\bar\beta}dz_\alpha\wedge d\bar z_{\beta}$. Define $n$ scalars $\rho_1$,...,$\rho_n$, by $$\frac{\det(g_{\alpha\bar\beta}+tR_{\alpha\bar\beta})}{\det(g_{\alpha\bar\beta})}=1+\sum_{k=1}^n\rho_k t^k$$ denote the scalar curvature of $M$, by $\rho$, then $\rho=2\rho_1$ and it is clear that $\rho_n=\frac{\det(R_{\alpha\bar\beta})}{\det(g_{\alpha\bar\beta})}$ Let $M$, is cohomologically Kähler-Einstein manifold, if $c_1(M)=a[\omega]$, then $$\int_M\rho_k\ast 1=(2\pi a)^k \binom{n}{k}\int_M\ast 1$$ where $\ast 1$ is the volume element.

A compact cohomological Kähler-Einstein manifold in Kähler-Einstein if the scalar curvature is constant.

Theorem. Let $M$ be an $n$- dimensional compact Kähler manifold , if $H^2(M,\mathbb R)\cong \mathbb R$, then $M$ is cohomological Kähler-Einstein manifold.

Theorem. Let $M$ be an $n$- dimensional compact Kähler manifold , and $\rho_k$ constant, $[Ric^k]=a[\omega^k]\in H^{2k}(M,\mathbb R)$, $\text{rank}(R_{\alpha\bar\beta})\geq k$ for some $k< n$ then $M$ is Kähler-Einstein manifold.

Theorem. Let $M$ be an $n$-dimensional compact cohomological Kähler-Einstein manifold, if there exists a $1\leq k\leq n$, such that $\rho_{k-1}$, and $\rho_{k}$ are positive constants, then $M$ is Kähler-Einstein manifold

https://www.jstor.org/stable/2040393

https://projecteuclid.org/euclid.jdg/1214432788