This is exercise 12.1 from Lectures on Symplectic Geometry by Ana Cannas da Silva. The task is to show that $$\omega := \frac{i}{2}\partial\overline{\partial}\log(|z|^2 + 1)$$ is a Kähler form on $\mathbb{C}^n$ (the Fubini-Study form). This results in showing that the function $\rho : \mathbb{C}^n \to \mathbb{R}$ defined by $$\rho(z) := \log(|z|^2 + 1)$$ is strictly plurisubharmonic. As a hint, we have:
$\omega$ is $U(n)$ invariant and thus it suffices to show positive definiteness along one direction.
Firts of all, I think that the invariance means $A^*\omega = \omega$, right? Where $A \in U(n)$. I do not quite see, how this should help showing that $$\frac{\partial^2 \rho}{\partial z_\mu \partial \overline{z}_\nu}(p)$$ is positive definite for every $p \in \mathbb{C}^n$. What I know is:
- How to show that above matrix is positive along one direction;
- How to show that the action of $U(n)$ on $\mathbb{S}^{2n - 1}$ is transitive;
- How to conclude that $\omega$ is a Kähler form.
So my question is only:
How does one formally show that if $\omega$ is invariant under $U(n)$, that then it suffices to show positive definiteness only along one direction?