  In this paper, we study the $SL_2(\mathbb{C})$ character variety of a
hyperbolic link in $S^3$. We analyze a special smooth projective variety $Y^h$
arising from some 1-dimensional irreducible slices on the character variety. We
prove that a natural symbol obtained from these 1-dimensional slices is a
torsion in $K_2({\mathbb C}(Y^h))$. By using the regulator map from $K_2$ to
the corresponding Deligne cohomology, we get some variation formulae on some
Zariski open subset of $Y^h$. From this we give some discussions on a possible
parametrized volume conjecture for both hyperbolic links and knots.
