  Let $X$ be a complex torus of dimension $g$ and $\hat{X}$ be the dual torus.
For any $g(g-1)/2$-tuple $\lambda$ of complex numbers of absolute value $1$, we
define a non-commutative complex torus $X_\lambda$ as a sheaf of algebras on a
real torus of dimension $g$. We prove that if all components of $\lambda$ are
roots of unity, then the category of coherent sheaves on $X_\lambda$ is abelian
and derived-equivalent to the category of coherent sheaves on $\hat{X}$ twisted
by an element of the Brauer group of $\hat{X}$ determined by $\lambda$.
