= 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 is
$$
\det\left(\frac{\partial^2u}{\partial z_j\partial\overline z_k}\right)_{j,k=1}^n=f(z),\qquad u\text{ plurisubharmonic}.
$$
The 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, $u_{z\overline z}=\tfrac14\Delta u$, so the equation reduces to a <Poisson equation>. On a <Kähler manifold>, its geometric version prescribes the volume form $(\omega+i\partial\overline\partial\varphi)^n$ of a positive metric. Different conventions for $i\partial\overline\partial$ change constants, so a determinant normalization should always accompany a coordinate formula.https://www3.nd.edu/~jdiller/teaching/archive/spring11_80380/blocki2.pdf[Zbigniew Błocki, The complex Monge-Ampère equation, local determinant formulation and Kähler geometry]{ref}
Back to article page
