Let $\omega = \frac{i}{2\pi} \sum dz_i \wedge d\bar{z_i}$ be the standard fundamental form on $\mathbb{C}^n$. Show that one can write $$\omega = \frac{i}{2\pi}\partial \bar{\partial} \varphi$$ for some positive function $\varphi$ and determine $\varphi$.
So I have two questions one minor detail related to the fundamental form and one for the actual problem.
- Why is there $\pi$ in the denominator of the scaling factor for the sum? This doesn't seem to be the "standard" fundamental form.
For the actual problem I started this by just considering $\mathbb{C}^1$ instead of $\mathbb{C}^n$. In $\mathbb{C}^1$ we want to find a positive function $\varphi$ such that $$\omega = \frac{i}{2\pi}\partial \bar{\partial} \varphi = \frac{i}{2\pi}\partial \left( \frac{\partial \varphi}{\partial \bar{z_1}} d \bar{z_1}\right) = \frac{i}{2\pi} \frac{\partial^2 \varphi}{\partial \bar{z_1}\partial z_1} d\bar{z_1} \wedge dz_1 = - \frac{i}{2\pi} \frac{\partial^2 \varphi}{\partial \bar{z_1}\partial z_1} dz_1 \wedge d\bar{z_1} $$ so $\varphi$ should be a positive function such that $$\frac{\partial^2 \varphi}{\partial \bar{z_1}\partial z_1} = -1$$ in order for us to get this to equal the standard fundamental form. I'm confused about this $$\frac{\partial^2 \varphi}{\partial \bar{z_1}\partial z_1}$$ term since I do not know what it should represent in this complex case. Is this even something that is defined as the complex partial derivatives are usually defined in terms of the expansion $x+iy$. What can we do here?