Complex solutions of Monge–Ampère equations

We describe a method to reduce partial differential equations of Monge–Ampère type in four variables to complex partial differential equations in two variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge–Ampère equations and the Plebanski equations.

► Complex solutions of Monge-Ampère equations are defined. ► Bieffective forms are associated with.► Geometric invariants are computed in complex dimension 4. ► Example of solutions are given for Special Lagrangian and Plebanski equations.