Complex Monge–Ampère operator

Definition

For a real-valued $C^2$ function $u$ on $U\subseteq\mathbb C^d$ , the complex Monge–Ampère operator is the determinant of its Levi matrix:

\operatorname{MA}_{\mathbb C}(u) =\det\!\left(\frac{\partial^2u} {\partial z_j\partial\bar z_k}\right).

If $u$ is plurisubharmonic, this determinant is nonnegative. Equivalently one packages it, up to the chosen normalization of $d^c$ , as the top-degree measure $(dd^cu)^d$ .

Nonsmooth extension

Bedford–Taylor theory defines $(dd^cu)^d$ for locally bounded plurisubharmonic functions by iterated products of positive currents. The result agrees with the smooth determinant measure and is continuous under appropriate monotone limits. For arbitrary unbounded PSH functions the operator is not automatically defined; an energy class or another domain of definition must be specified.

Real and quaternionic analogues

The real Monge–Ampère operator takes the determinant of the real Hessian. Replacing the complex Hessian by the quaternionic Hessian and the ordinary determinant by the Moore determinant gives the quaternionic Monge–Ampère measure.

References

Eric Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions,” Acta Mathematica 149 (1982), 1–40. DOI record.
Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter III.