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

$SL_2(\mathbb{C})$-Character Variety of a Hyperbolic Link and Regulator