= Kähler-Einstein metric
{c}
{title2=$\operatorname{Ric}(g)=\lambda g$}
A Kähler-Einstein metric is a <Kähler metric> $g$ whose <Ricci tensor> satisfies
$$
\operatorname{Ric}(g)=\lambda g
$$
for a real constant $\lambda$. Equivalently, its <Ricci form> satisfies $\rho=\lambda\omega$, where $\omega$ is the <Kähler form>. It combines the compatibility of complex and Riemannian geometry with an Einstein curvature condition.
In holomorphic coordinates, take $\omega=i\sum g_{j\bar k}\,dz^j\wedge d\bar z^k$. The <Ricci form> is $\rho=-i\partial\bar\partial\log\det(g_{j\bar k})$. A constant positive <Hermitian matrix> therefore gives $\rho=0$: the flat metric on $\mathbb C^n$, and its quotient by a full lattice, are Ricci-flat examples. The <Fubini-Study metric> on <Complex projective space> gives a positive-$\lambda$ example, with the numerical value depending on the metric normalization.
The equation is a nonlinear condition on a local <Kähler potential>, through the logarithmic determinant of its complex <Hessian>. This is why <complex Monge-Ampère equations> enter existence questions. Not every <Kähler metric> satisfies the Einstein condition.<Lectures on Kähler Geometry by Andrei Moroianu>[Sections 11 and 13, Ricci form and Kähler-Einstein condition]{ref}
Back to article page
