Hermitian matrices. For the $T_{\nu}$ function we fix our volume form $\nu$ as the standard Fubini-Study form $$\label{nu form}
    \nu = \sqrt{-1}\partial\overline{\partial}\log(1+|z|^2)
    = \frac{\sqrt{-1}}{(1+|z|^2)^2} \mathrm{d}z\wedge\mathrm{d}\overline{z}.$$ In the case of the $T_K$ map we note that $K = O(-2)$, hence $L^k = K^{-p}$ precisely when $k=2p.$

We simplify further by considering only those metrics invariant under the $S^1$ action $z \mapsto e^{i\theta}z$ on the Riemann sphere. This restricts our attention to [*diagonal*]{} positive definite Hermitian $(k+1)\times(k+1)$ matrices $G$. We will suppose $G$ has entries $a_0^{-1}, a_1^{-1}, \ldots, a_k^{-1}$ – taking inverses simplifies later computations – and we will use the notation $$G = \left( a_0, a_1, \ldots, a_k \right)$$ to denote this metric. Each of $T, T_{\nu},$ and $T_K$ is a function of $(a_0, a_1, \ldots, a_k)$, and in the remainder of this section we write them down explicitly.

We begin with $T.$ Taking $G$ as above we can pick the orthonormal basis $\{s_i = \sqrt{a_i}\,z^i,\ i=0,\ldots,k\}.$ Then $$h = FS(G) = \left( \sum a_i |z|^{2i} \right)^{-1}$$ and we calculate $$\omega_h = \sqrt{-1}\partial\overline{\partial}\log\left(\sum a_i|z|^{2i} \right)
        = \sqrt{-1} \frac{\sum_{i>j}a_ia_j(i-j)^2|z|^{2(i+j-1)}}{\left(\sum a_i|z|^{2i}\right)^2}
            \mathrm{d}z\wedge\mathrm{d}\overline{z}.$$ Write $T(a_0, \ldots, a_k) = (\tilde{a}_0, \ldots, \tilde{a}_k).$ Then $$\tilde{a}_q^{-1} = R \int_{\mathbb{C}} |z|^{2q} h \omega_h,$$ where $R = (k+1)/\mathrm{Vol}(X, \omega_h)$. 