Nobuki Okuda

Fourier--Mukai transforms for non-commutative complex tori

Let $X$ be a complex torus of dimension $g$ and $\hat{X}$ be the dual torus. For any $g(g-1)/2$-tuple $λ$ of complex numbers of absolute value $1$, we define a non-commutative complex torus $X_λ$ as a sheaf of algebras on a real torus of dimension $g$. We prove that if all components of $λ$ are roots of unity, then the category of coherent sheaves on $X_λ$ 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 $λ$.