Definition: the Fubini-Study metric $g_{FB}$ on $\mathbb{CP}^n$ is the only metric which makes the projection $\pi:(\mathbb{S}^{2n+1},g)\to(\mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is the standard metric)
I'm trying to deduce the coefficients of $g_{FB}$ in charts, i.e.: \begin{align*} g_{FB}\left(\frac{\partial}{\partial x_\alpha},\frac{\partial}{\partial x_\beta}\right)=g_{FB}\left(\frac{\partial}{\partial y_\alpha},\frac{\partial}{\partial y_\beta}\right)&=\frac{1}{(1+|z|^2)^2}((1+|z|^2)\delta_{\alpha\beta}-(x_\alpha x_\beta+y_\alpha y_\beta))\\ g_{FB}\left(\frac{\partial}{\partial x_\alpha},\frac{\partial}{\partial y_\beta}\right)&=\frac{-1}{(1+|z|^2)^2}(x_\alpha y_\beta-y_\alpha x_\beta) \end{align*}
Since $x_\alpha x_\beta+y_\alpha y_\beta=\text{Re}(z_\alpha\overline{z}_\beta)$ and $-(x_\alpha y_\beta-y_\alpha x_\beta)=\text{Im}(z_\alpha\overline{z}_\beta)$, I thought maybe it would be a good idea to use complex variables, but I don't know how to do that from the condition that $\pi$ is a Riemannian submersion.
Any suggestions?
It's indeed very natural to use the hermitian geometry structure here. Let $e_0;e_1,\dots e_n$ be a unitary frame at the point $[Z]=[e_0]\in\Bbb P^n$. Then $$de_0 = \omega_{0\bar 0}e_0 + \sum \omega_{0\bar j}e_j$$ and the hermitian metric on $\Bbb P^n$ is given by $\sum |\omega_{0\bar j}|^2 = \sum \omega_{0\bar j}\overline{\omega_{0\bar j}}$. In one of your charts, say $Z_0\ne 0$, we take coordinates by setting $Z=(1,z)$ and $e_0 = Z/\|Z\|$.
Now, note that $\omega_{0\bar 0} = i\,d\theta$ where $e^{i\theta}$ gives the fiber of your Riemannian submersion. Moreover, denoting the hermitian inner product by $(\cdot,\cdot)$, $$\sum |\omega_{0\bar j}|^2 = (de_0,de_0) - |\omega_{0\bar 0}|^2 = (de_0,de_0)-|(de_0,e_0)|^2.$$ Substituting $e_0 = \dfrac{(1,z)}{\|(1,z)\|}$ and differentiating appropriately, you'll get your desired formula for the metric as a $2$-tensor in terms of $dz_j$ and $d\bar z_j$.