The set of all orthogonal complex structures on the flat 6-tori

In [2], Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori TR2n for any n≥3. We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.