terms $a_{i}^{-1}$. Such a matrix corresponds to an (algebraic) metric invariant under the torus $\Lambda_n = (S^1)^n$ action $z_l \mapsto e^{i\theta_l}z_l,\ l=1,\ldots,n$.

An orthonormal basis, according to $G$, is given by $$\left\{\sqrt{a_{i}}\,w_i, i=1, \ldots, N \right\}.$$ Then in terms of the $a_i$’s the matrix ${T_{\nu}}(G)$ will have diagonal entries $\tilde{a}_i^{-1}$ equal to $$\| w_i \|_{\mathrm{Hilb}_{\nu}} =
    N n! \left( \frac{\sqrt{-1}}{2\pi} \right)^n \int_{{\mathbb{C}}^n} \frac{
        |w_i|^2 dz_1\wedge d\overline{z}_1\wedge\cdots \wedge dz\wedge d\overline{z}_n
        }{
        \left( \sum_{p=1}^N a_{p} |w_p|^2 \right) \left( 1+\sum_{q=1}^n |z_q|^2 \right)^{n+1}
        }.$$ Changing to polar coordinates $z_j = r_j \exp\left(\sqrt{-1}\ \theta_j\right)$ and substituting $x_j=r_j^2$ we get $$\label{eqn: CPn integral}
    T(G)_{ii}=\tilde{a}_{i}^{-1} =
    N n! \int_0^{\infty}\hspace{-10pt}\cdots\!\!\int_0^{\infty}
    \frac{w_i(x) dx_1\cdots dx_n}{
        \left(\sum_{p=1}^N a_p w_p(x) \right) \left(1 + \sum_{q=1}^n x_q \right)^{n+1}
    }$$ where $w(x)$ denotes the monomial $w$ with the substitutions $x_k = z_k, k=1, \ldots, n$.

Asymptotic behavior in higher dimensions
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Let us consider the asymptotic behavior of ${T_{\nu}}$. Recall (see section \[subsection: Asymptotic behavior\]) that in the case of $n=1$, i.e. when $X={\mathbb{CP}}^1$, we defined $$\sigma_{T_{\nu},k} := \lim_{r \rightarrow \infty}
        \frac{
            \mathrm{dist}({T_{\nu}}^{(r+1)}(G),\ {T_{\nu}}^{(\infty)}(G))
        }{
            \mathrm{dist}({T_{\nu}}^{(r)}(G),\ {T_{\nu}}^{(\infty)}(G))
        }.$$ This value depends on whether or not the initial metric $G$ is invariant under the inversion map $z \mapsto z^{-1}$, or equivalently in homogeneous coordinates $Z_0 \leftrightarrow 