Fourier--Mukai transforms for non-commutative complex tori

TL;DR

This paper extends Fourier--Mukai transforms to non-commutative complex tori, showing derived equivalences with dual tori under certain conditions on the twisting parameters.

Contribution

It introduces non-commutative complex tori defined via sheaves of algebras and establishes derived equivalences with dual tori when parameters are roots of unity.

Findings

Category of coherent sheaves on non-commutative tori is abelian.

Derived equivalence with twisted coherent sheaves on dual torus.

Conditions for equivalence involve parameters being roots of unity.

Abstract

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 $\lambda$.