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arXiv · 2506.18179

Exotic hypercomplex structures on a torus do not exist

Alberto Pipitone Federico·Misha Verbitsky
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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.

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Alberto Pipitone Federico, Misha Verbitsky. 2025-06-22. Exotic hypercomplex structures on a torus do not exist. https://arxiv.org/abs/2506.18179

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