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

TL;DR

This paper classifies all orthogonal complex structures on flat 6-tori, showing they are either standard complex tori or the non-standard BSV-tori, thus solving a classification problem in complex dimension three.

Contribution

It proves that on flat 6-tori, all orthogonal complex structures are either standard or BSV-tori, completing the classification of such structures in complex dimension three.

Findings

All orthogonal complex structures on flat 6-tori are classified as either complex tori or BSV-tori.

The classification solves the problem for compact Hermitian manifolds with flat Riemannian connection in complex dimension three.

Abstract

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq 3$. 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.