Complex Monge–Ampère operator

Remarks

The constant $4^n n!$ depends on the normalisation of $d^c$; many authors use $d^c = \frac{i}{4\pi}(\bar\partial - \partial)$ or write simply $\det\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$. If $u$ is plurisubharmonic, the complex Hessian is positive semidefinite, so $(dd^c u)^n$ is a positive measure. Example: for $u(z) = \abs{z}^2 = \sum_j \abs{z_j}^2$ the complex Hessian is the identity, so $(dd^c u)^n = 4^n n! \, dV$. For $n = 1$, $dd^c u = 4 \, \frac{\partial^2 u}{\partial z \partial \bar z} \, dV = \Delta u \, dV$, the Laplacian. Non-example of positivity: for $u = \abs{z_1}^2 - \abs{z_2}^2$ on $\C^2$ the determinant is $-1$, so $(dd^c u)^2$ is a negative measure; this $u$ is not plurisubharmonic. The operator is nonlinear for $n \ge 2$. The extension of $(dd^c u)^n$ to locally bounded plurisubharmonic functions, where $u$ need not be differentiable, is the Bedford–Taylor theory.