  Extending the usual $\mathbf{C}^{\ast r}$ actions of toric manifolds by
allowing asymmetries between the various $\mathbf{C}^{\ast}$ factors, we build
a class of non commutative (NC) toric varieties $\mathcal{V}%_{d+1}^{(nc)}$. We
construct NC complex $d$ dimension Calabi-Yau manifolds embedded in
$\mathcal{V}_{d+1}^{(nc)}$ by using the algebraic geometry method. Realizations
of NC $\mathbf{C}^{\ast r}$ toric group are given in presence and absence of
quantum symmetries and for both cases of discrete or continuous spectrums. We
also derive the constraint eqs for NC Calabi-Yau backgrounds
$\mathcal{M}_{d}^{nc}$ embedded in $\mathcal{V}_{d+1}^{nc}$ and work out their
solutions. The latters depend on the Calabi-Yau condition $%
\sum_{i}q_{i}^{a}=0$, $q_{i}^{a}$ being the charges of $\mathbf{C}^{\ast r}$% ;
but also on the toric data ${q_{i}^{a},\nu_{i}^{A};p_{I}^{\alpha},\nu
_{iA}^{\ast}} $ of the polygons associated to $\mathcal{V}%_{d+1}$. Moreover,
we study fractional $D$ branes at singularities and show that, due to the
complete reducibility property of $\mathbf{C}^{\ast r}$ group representations,
there is an infinite number of fractional $D$ branes. We also give the
generalized Berenstein and Leigh quiver diagrams for discrete and continuous
$\mathbf{C}^{\ast r}$ representation spectrums. An illustrating example is
presented.
