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The complex Monge-Ampère equation prescribes the determinant of the complex Hessian of a real-valued function. On a domain in ${\mathbb{C}}^n$, one local normalization isThe positivity condition means that the complex Hessian is positive semidefinite; where it is positive definite the equation is elliptic. The determinant is nonlinear when $n>1$, unlike the Laplace operator, which takes a trace.

For $u(z)={\sum }_j|z_j{|}^2$, the complex Hessian is the identity and $f=1$. In one complex dimension, $u_{z\overline{z}}=\frac{1}{4}\Delta u$, so the equation reduces to a Poisson equation. On a Kähler manifold, its geometric version prescribes the volume form $(\omega +i\partial \overline{\partial }\phi {)}^n$ of a positive metric. Different conventions for $i\partial \overline{\partial }$ change constants, so a determinant normalization should always accompany a coordinate formula.^[ref]

A Kähler-Einstein metric is a Kähler metric $g$ whose Ricci tensor satisfiesfor 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 $\omega =i\sum g_{j\bar{k}}d{z}^j\wedge d{\bar{z}}^k$. 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.^[ref]