  Let $M$ be a hyperbolizable $3$-manifold with boundary, and let $\chi_0(M)$
be a component of the $PSL_2\mathbb{C}$-character variety of $M$ that contains
the convex co-compact characters. We show that the peripheral map
$i_*:\chi_0(M)\rightarrow\chi(\partial M)$ to the character variety of
$\partial M$ is a birational isomorphism with its image, and in particular is
generically a one-to-one map. This generalizes work of Dunfield (one cusped
hyperbolic $3$-manifolds) and Klaff-Tillmann (finite volume hyperbolic
$3$-manifolds). We use the Bonahon-Schl\"afli formula and volume rigidity of
discrete co-compact representations.
