  Using the algebraic geometry method of Berenstein and Leigh (BL),
hep-th/0009209 and hep-th/0105229), and considering singular toric varieties
${\cal V}_{d+1}$ with NC irrational torus fibration, we construct NC extensions
${\cal M}_{d}^{(nc)}$ of complex d dimension Calabi-Yau (CY) manifolds embedded
in ${\cal V}_{d+1}^{(nc)}$. We give realizations of the NC $\mathbf{C}^{\ast
r}$ toric group, derive the constraint eqs for NC Calabi-Yau (NCCY) manifolds
${\cal M}^{nc}_d$ embedded in ${\cal V}_{d+1}^{nc}$ and work out solutions for
their generators. 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 non compact fractional branes
at fixed points of the NC toric group.
