# complex Hessian matrix

Suppose that $f\colon{\mathbb{C}}^{n}\to\mathbb{C}$ be twice differentiable and let

 $\frac{\partial}{\partial z_{k}}:=\frac{1}{2}\left(\frac{\partial}{\partial x_{% k}}-i\frac{\partial}{\partial y_{k}}\right)\quad\text{ and }\quad\frac{% \partial}{\partial\bar{z}_{k}}:=\frac{1}{2}\left(\frac{\partial}{\partial x_{k% }}+i\frac{\partial}{\partial y_{k}}\right).$

Then the is the matrix

 $\begin{bmatrix}\frac{\partial^{2}f}{\partial z_{1}\partial\bar{z}_{1}}&\frac{% \partial^{2}f}{\partial z_{1}\partial\bar{z}_{2}}&\ldots&\frac{\partial^{2}f}{% \partial z_{1}\partial\bar{z}_{n}}\\ \frac{\partial^{2}f}{\partial z_{2}\partial\bar{z}_{1}}&\frac{\partial^{2}f}{% \partial z_{2}\partial\bar{z}_{2}}&\ldots&\frac{\partial^{2}f}{\partial z_{2}% \partial\bar{z}_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial^{2}f}{\partial z_{n}\partial\bar{z}_{1}}&\frac{\partial^{2}f}{% \partial z_{n}\partial\bar{z}_{2}}&\ldots&\frac{\partial^{2}f}{\partial z_{n}% \partial\bar{z}_{n}}\end{bmatrix}.$

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).

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title complex Hessian matrix ComplexHessianMatrix 2013-03-22 14:31:16 2013-03-22 14:31:16 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32-00 HessianMatrix