numerical findings in the cases of $n=2$ and $n=3$. The itereated integrals (equation \[eqn: CPn integral\]) grow in computational complexity quickly with increasing $n$.

We start with a metric $G$ which is torus-invariant, but otherwise ‘random’ in the sense that it is not invariant under any permutation of the homogeneous coordinates. We tabulate approximate numerical values for the asymptotic constant $\sigma$ here, all computed starting with ‘random’ (but torus-invariant) metrics:

   $\sigma$   $k=2$    3      4      5
  ---------- ------- ------ ------ ------
    $n=2$     0.40    0.50   0.57   0.63
      3       0.33    0.43   0.50   0.56

For the moment let us just note that the above values apparently follow the pattern: $$\label{eqn: sigma for Tv generic case}
    \sigma = \frac{k}{k+n+1}.$$ When $n=1$, the fundamental case which we considered, this formula specializes to (\[eqn: sigma for CP1\]).

The non-generic case, when $G$ might be invariant under a permutation of the homogeneous coordinate variables we find simple behavior:

-   If there is no fixed-point-free permutation of the homogeneous coordinate variables under which $G$ is invariant then $\sigma$ is the same as computed in the asymmetric case.

-   Otherwise suppose $G$ is invariant under some fixed-point-free permutation of the homogeneous coordinate variables. Then we get new values for $\sigma$, 