Complex Monge–Ampère operator

Summary

The complex Monge–Ampère operator sends a function $u$ on an open subset of $\C^n$ to the determinant of its complex Hessian. The complex Hessian is the matrix $\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$, and the determinant is viewed as a measure. In the language of forms it is the $n$-th exterior power $(dd^c u)^n$. For $n = 1$ it is the Laplacian, so it is the nonlinear analogue of the Laplacian in pluripotential theory. It is the operator of the complex Monge–Ampère equation, which appears in the construction of Kähler–Einstein metrics and in the study of plurisubharmonic functions.