Kähler–Einstein metric

Remarks

Example: the flat metric on a complex torus $\C^n / \Lambda$, for a lattice $\Lambda$ of rank $2n$. It is Kähler with constant coefficients, its curvature vanishes, and so $\operatorname{Ric} = 0 = 0 \cdot g$. The Fubini–Study metric on $\C P^n$ is Kähler–Einstein with $\lambda > 0$, and the Bergman metric on the unit ball of $\C^n$ is Kähler–Einstein with $\lambda < 0$.

Non-example: on $\C P^1 \times \Sigma$, with $\Sigma$ a compact Riemann surface of genus $2$, take the product of a round metric of curvature $1$ and a metric of curvature $-1$. It is Kähler, but $\operatorname{Ric}$ equals $g$ on the first factor and $-g$ on the second, so no single $\lambda$ works.

On a compact Kähler manifold the Ricci form represents $2\pi c_1(M)$ in de Rham cohomology. So a Kähler–Einstein metric with constant $\lambda$ forces $2\pi c_1(M) = \lambda [\omega]$: the first Chern class is negative, zero or positive according to the sign of $\lambda$. A Kähler–Einstein metric is an Einstein metric in the Riemannian sense.