  We organize the quantum hyperbolic invariants (QHI) of $3$-manifolds into
sequences of rational functions indexed by the odd integers $N\geq 3$ and
defined on moduli spaces of geometric structures refining the character
varieties. In the case of one-cusped hyperbolic $3$-manifolds $M$ we generalize
the QHI and get rational functions $\mathcal{H}_N^{h_f,h_c,k_c}$ depending on a
finite set of cohomological data $(h_f,h_c,k_c)$ called {\it weights}. These
functions are regular on a determined Abelian covering of degree $N^2$ of a
Zariski open subset, canonically associated to $M$, of the geometric component
of the variety of augmented $PSL(2,\mathbb{C})$-characters of $M$. New
combinatorial ingredients are a weak version of branchings which exists on
every triangulation, and state sums over weakly branched triangulations,
including a sign correction which eventually fixes the sign ambiguity of the
QHI. We describe in detail the invariants of three cusped manifolds, and
present the results of numerical computations showing that the functions
$\mathcal{H}_N^{h_f,h_c,k_c}$ depend on the weights as $N\rightarrow + \infty$,
and recover the volume for some specific choices of the weights.
