Complex Monge-Ampère equation

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]