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

Abstract

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

14 pages

References in corpus (1)

• On Curvature Tensors of Hermitian Manifolds