complex Hessian matrix

Suppose that $f:{ℂ}^n\to ℂ$ be twice differentiableand let

$$\frac{\partial }{\partial z_k}:=\frac{1}{2}\left(\frac{\partial }{\partial x_k}-i\frac{\partial }{\partial y_k}\right)\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}\frac{\partial }{\partial {\overline{z}}_k}:=\frac{1}{2}\left(\frac{\partial }{\partial x_k}+i\frac{\partial }{\partial y_k}\right).$$

Then the is the matrix

When applied to tangent vectors of the zero set of $f$,it is called the Levi form and used to define a Levipseudoconvex point of a boundary of a domain. Note that the matrix is not the same as the (real) Hessian. A twice continuouslydifferentiable real valuedfunction with apositive semidefinite real Hessian matrix at every point is convex, but a function withpositive semidefinite matrix at every point isplurisubharmonic (since it’scontinuous it’s also called a pseudoconvex function).