Hessian for Complex

Question

I have tried to find the Hessian matrix for complex function. first derivative of $$f$$ with respect to x and then for y. Next to that we find $$f_{xx}, f_{xy}, f_{yx}, f_{yy}$$ $$ \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \\ \end{bmatrix} $$ Could you tell me if it is Ok, please?

Answer 1

For a function $f = f(z_1, \ldots, z_n)$ of several complex variables, the usual meaning of the Complex Hessian is the matrix $$ \begin{bmatrix} \dfrac{\partial^2 f}{\partial z_1 \partial \bar z_1} & \dfrac{\partial^2 f}{\partial z_1 \partial \bar z_2} & \ldots & \dfrac{\partial^2 f}{\partial z_1 \partial \bar z_n} \\ \dfrac{\partial^2 f}{\partial z_2 \partial \bar z_1} & \dfrac{\partial^2 f}{\partial z_2 \partial \bar z_2} & \ldots & \dfrac{\partial^2 f}{\partial z_1 \partial \bar z_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial^2 f}{\partial z_n \partial \bar z_1} & \dfrac{\partial^2 f}{\partial z_n \partial \bar z_2} & \ldots & \dfrac{\partial^2 f}{\partial z_n \partial \bar z_n} \end{bmatrix} $$ where $\dfrac{\partial}{\partial z_j} = \dfrac12\Big( \dfrac{\partial}{\partial x_j} - i\,\dfrac{\partial}{\partial y_j}\Big)$ and $\dfrac{\partial}{\partial \bar z_j} = \dfrac12\Big( \dfrac{\partial}{\partial x_j} + i\,\dfrac{\partial}{\partial y_j}\Big)$ are Wirtinger derivatives.

For example if $u$ is (a real-valued $C^2$) function, then $u$ is plurisubharmonic if and only if the complex Hessian of $u$ is positive semi-definite.

The determinant of the complex Hessian (suitably normalized) is known as the complex Monge-Ampère operator, a central object of study in pluripotential theory.

Answer 2

To add to mrf's answer. The complex Hessian matrix is only "one half" of the real Hessian. Clearly, $f$ still has a regular real Hessian. Let's do it in $n=1$ since it's much easier then. The function $f \colon \mathbb C \to \mathbb C$ can be then taken as a function of $x,y$ or of $z$ and $\bar{z}$. Going from $x,y$ to $z,\bar{z}$ is a linear change of variables as $x = \frac{z+\bar{z}}2$ and $y = \frac{z-\bar{z}}{2i}$. Then the real hessian in these variables is $$ \begin{bmatrix} \frac{\partial^2 f}{\partial z^2} & \frac{\partial^2 f}{\partial z \partial \bar{z}} \\ \frac{\partial^2 f}{\partial \bar{z} \partial z} & \frac{\partial^2 f}{\partial \bar{z}^2} \end{bmatrix} $$ The entry $\frac{\partial^2 f}{\partial z \partial \bar{z}}$ is the so called complex hessian, the entry $\frac{\partial^2 f}{\partial z^2}$ is the holomorphic hessian. (You also get the what I'd call the "anti-holomorphic hessian" in the lower right)

In several variables, it is the same thing, but these entries become submatrices.

The "complex hessian" is the bit that is "preserved" by holomorphic changes of coordinates. On the other hand, the "holomorphic hessian" is really the hessian you'd look at if looking at holomorphic functions. It all depends on what you want to do with it.