complex Hessian matrix

Suppose that $f:{ℂ}^n\to ℂ$ be twice differentiable and 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 Levi pseudoconvex point of a boundary of a domain. Note that the matrix is not the same as the (real) Hessian. A twice continuously differentiable real valued function with a positive semidefinite real Hessian matrix at every point is convex, but a function with positive semidefinite matrix at every point is plurisubharmonic (since it’s continuous it’s also called a pseudoconvex function).

Title	complex Hessian matrix
Canonical name	ComplexHessianMatrix
Date of creation	2013-03-22 14:31:16
Last modified on	2013-03-22 14:31:16
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Definition
Classification	msc 32-00
Related topic	HessianMatrix