Calculating the inverse components of the Fubini-Study Metric. In coordinates $(z_1, \dotsc, z_n)$, the Fubini-Study metric, can be written as 
$ds^2 = \frac{(1 + z_i\overline{z}^i)dz_jd\overline{z}^j - \overline{z}^jz_idz_jd\overline{z}^i}{(1 + z_i\overline{z}^i)^2}$
where there is an implicit sum over $i$ and $j$ ranging from $1$ to $n$. From this I can calculate the metric components $g_{kl}$ where $k,l \in \{1, \dotsc, n, \overline{1}, \dotsc, \overline{n}\}$.
These are the metric components I've calculated:
$g_{kl} = g_{\overline{k}\overline{l}} = 0$ for $k, l \in \{1, \dotsc, n\} k \neq l$.
$g_{k\overline{l}} = -\frac{\overline{z}^lz_k}{(1 + \sum_{i = 1}^nz_i\overline{z}^i)^2}$ for $k, l \in \{1, \dotsc, n\}$.
$g_{\overline{k}l} = (g_{k\overline{l}})^*$ for $k, l \in \{1, \dotsc, n\} $.
$g_{kk} = \frac{1}{(1 + \sum_{i =1}^nz_i\overline{z}^i)}$ for $k \in \{1, \dotsc, n\}$.
$g_{\overline{k}\overline{k}} = 0$ for $k \in \{1, \dotsc, n\}$.
My question is: what are the inverse components? Is there an easy way to calculate them? 
 A: The Fubini-Study metric is a metric on $\mathbb{CP}^n$, so where looking at coordinates $Z$ which are equivalence classes $Z = [Z_0,\dots,Z_n]$, however to make our lives simpler, we look at the local affine coordinates in the coordinate patch traditionally denoted $U_0 = \{Z_0 \neq 0\}$ and then normalise the coordinate $[Z_0,\dots,Z_n] \sim [1,z_1,\dots,z_n]$. This gives the line element $ds^2$ which you've written above.
One way to calculate the inverse metric tensor elements is to calculate the matrix constructed by the coefficients, i.e. $[g_{i\bar{j}}]$. This is a square matrix and we can calculate it's inverse. The coefficients of this inverse will be, by definition the inverse components. I used the formula for the Fubini-Study metric coefficients from here.
$$g_{i\bar{j}} = \frac{(1+|z|^2) \delta_{i\bar{j}} - \bar{z}_i z_j}{(1+|z|^2)^2}$$
I wrote a quick and dirty Mathematica script which generates a matrix from these coefficients, so that I could calculate the inverse.
Delta[i_,j_] := If[i==j,1,0];
z = {a,b,c,d};
HermiteNorm[z_]:= Sum[z[[i]]*Conjugate[z[[i]]],{i,1,Length[z]}]
h[i_,j_] := ((1 + HermiteNorm[z])*Delta[i,j] - Conjugate[z[[i]]]*z[[j]]);
H = (1/(1 + HermiteNorm[z])^2)*Array[h,{Length[z],Length[z]}];
G = MatrixPower[H,-1];
Print[G//MatrixForm//Simplify]

This will generate a 4x4 matrix for the coordinate $z = (a,b,c,d)$, which gives you an example of the inverse matrix coefficients. Admittedly, there should be a way to calculate the coefficients directly, however the underlying result comes from taking the inverse of a matrix. The resultant matrix gives the inverse coefficients by the following formula:
$$g^{i\bar{j}} = (1 + |z|^2)(\delta_{i\bar{j}}(1 + z_j \bar{z}_i) + (1 - \delta_{i\bar{j}})z_j\bar{z}_i)$$
