Consider the projective space $\mathbb{P}^n = \mathbb{P}(\mathbb{C}^{n+1})$ and the open sets $$U_i = \{ [Z_0:Z_1:\ldots:Z_n] \in \mathbb{P}^n: \ Z_i \neq 0\}$$ for $i = 0 \ldots n$. These sets cover $\mathbb{P}^n$ and we can use each $U_i$ to define a coordinate system (affine coordinates) $(U_i, \varphi_i)$ such that $$\varphi_i([Z_0:\ldots:Z_n]) = \left( \frac{Z_0}{Z_i}, \ldots, \frac{Z_{i-1}}{Z_i}, \frac{Z_{i+1}}{Z_i}, \ldots, \frac{Z_n}{Z_i}\right) = (z_1, \ldots, z_n).$$
With respect to this coordinate system the Fubini-Study metric is given by the (1,1)-form $$\omega = \frac{i}{2\pi}\sum_{i,j=1}^n h_{ij} dz_i \wedge d\overline{z}_j,$$ where $$h_{ij} = \frac{(1+\|z\|^2)\delta_{ij} - \overline{z}_i z_j}{(1+\|z\|^2)^2}$$ and $z = (z_1, \ldots, z_n)$.
Now consider another possible coordinate system. We have the open sets $\mathbb{A}_Z = \{ [Y+Z] \in \mathbb{P}^n: \ Y \in Z^\perp\}$ for each $Z \in \mathbb{C}^{n+1}$ (we are considering $\mathbb{C}^{n+1}$ with the usual Hermitian inner product to define $Z^\perp$). These sets also cover $\mathbb{P}^n$ and we can use each $\mathbb{A}_Z$ to define the new coordinate system $(\mathbb{A}_Z, \psi_Z)$ such that $$\psi_Z([W_0:\ldots:W_n]) = \frac{\|Z\|^2}{\langle W, Z \rangle} W - Z,$$ where $W = (W_0, \ldots, W_n) \in \mathbb{C}^{n+1}$. Although $\text{Im}(\psi_Z) \subset \mathbb{C}^{n+1}$, it is a subspace of $\mathbb{C}^{n+1}$ of dimension $n$.
This coordinate system has a very nice geometric interpretation and is as natural as the affine coordinate system. But in all texts I've seen everybody only talks about Fubini-Study metric in affine coordinates. How is the computation of the Fubini-Study metric is this particular coordinate system?
Any recommendation of reading is welcome. Specially if this specific question is addressed! I already read this topic on Griffiths-Harris (Principles of Algebraic Geometry) and Daniel Huybrechts (Complex Geometry). Both texts make the same thing: they introduce the Fubini-Study metric in affine coordinates. I also gave a look at the Wikipedia article but it didn't help me.
PS: In Griffiths-Harris they say the form $\omega$ is invariant by unitary action. This means $\omega_{Z}(u,v) = \omega_{UZ}(Uu, Uv)$ for all $Z \in \mathbb{C}^{n+1}\backslash\{0\}$, all $U \in \mathbb{C}^{(n+1) \times (n+1)}$ unitary and all tangent vectors $u,v \in z^\perp \cong T_{[Z]}\mathbb{P}^n$ ?
Thank you for your help.
I got very helpful comments thanks to AlanMuniz, but not an official answer. So I'll try to give mine and I hope this can be helpful to someone. Also, if there is something wrong I hope someone can point it out to me.
Following Griffiths-Harris terminology, the map $W\backslash\{0\} \in \mathbb{C}^{n+1} \mapsto L_Z(W) = \displaystyle \frac{\|Z\|^2}{\langle W,Z \rangle}W$ is not a chart but a lifting, i.e., a holomorphic section of the total space $\mathbb{C}^{n+1}$ over the base $\mathbb{P}^n$. Since $\omega_Z$ is invariant by unitary action, we have that $$\omega_{Z}(u,v) = \omega_{UZ}(Uu, Uv)$$ for every unitary matrix $U \in \mathbb{C}^{(n+1) \times (n+1)}$ and every tangent vectors $u,v \in Z^\perp \cong T_{[Z]}\mathbb{P}^n$. Associated to this change we have the lifting $L_{UZ}(W) = \displaystyle \frac{\|UZ\|^2}{\langle W,UZ \rangle}W$, so $$\omega_{UZ} = \frac{i}{2\pi} \partial \overline{\partial}\log\left( \|L_{UZ}\|^2 \right) = \frac{i}{2\pi} \partial \overline{\partial}\log\left( \left\|\frac{\|UZ\|^2}{\langle W,UZ \rangle}W\right\|^2 \right) = \frac{i}{2\pi} \partial \overline{\partial}\log\left( 1+\|z\|^2 \right),$$ where $z = \left( \frac{(UZ)_1}{(UZ)_0}, \ldots, \frac{(UZ)_n}{(UZ)_0} \right)$ and $(UZ)_i$ denotes the $i$-th coordinate of $UZ$. Now we are working in affine coordinates. In this case we have that $$\omega_Z = \frac{i}{2\pi} \sum_{i,j=1}^n h_{ij} dz_i \wedge d\overline{z}_j,$$ where $$h_{ij} = \frac{(1+\|z\|^2) \delta_{ij} - \overline{z}_i z_j}{(1+\|z\|)^2}.$$
Extra: We can take one step further. In particular, there is a matrix $U$ such that $UZ = e_0 = (1,0,\ldots,0)$. Note that $h_{ij} = \delta_{ij}$. Therefore, $$\omega_{e_0} = \frac{i}{2\pi} \sum_{j=1}^n dz_j \wedge d\overline{z}_j.$$
These results agree with the text of Griffiths-Harris.