  Under the assumption of asymptotic relative Chow-stability for polarized
algebraic manifolds $(M, L)$, a series of weighted balanced metrics $\omega_m$,
$m \gg 1$, called polybalanced metrics, are obtained from complete linear
systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to
\infty$ will be studied.
