Exotic hypercomplex structures on a torus do not exist

TL;DR

This paper proves that exotic hypercomplex structures, which are hypercomplex structures on tori that are not hyperkahler, do not exist by analyzing the flatness of the Obata connection and classifying affine structures on tori.

Contribution

It demonstrates the non-existence of exotic hypercomplex structures on tori through classification of flat affine structures and analysis of the Obata connection.

Findings

Obata connection for exotic structures is flat

Complete flat affine structures on tori are classified

Exotic hypercomplex structures on tori do not exist

Abstract

A hypercomplex manifold is a manifold with three complex structures satisfying quaternionic relations. Such a manifold admits a unique torsion-free connection preserving the quaternionic action, called the Obata connection. A compact Kahler manifold admitting a hypercomplex structure always admits a hyperkahler structure as well; however, it is not obvious whether the original hypercomplex structure is hyperkahler. A non-hyperkahler hypercomplex structure on a Kahler manifold is called exotic. We show that the Obata connection for an exotic hypercomplex structure on a torus is flat and classify complete flat affine structures on real tori. We use this classification to prove that exotic hypercomplex structures do not exist.