Exotic hypercomplex structures on a torus do not exist

2025/06/22 by Federico, Alberto Pipitone, Verbitsky, Misha
#53C26 #53C28 #53C55 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.18179

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.

Citations

• The twistor space of a compact hypercomplex manifold is never Moishezon
• Divisorial properties and special metrics on hypercomplex twistor spaces
• The deformations of flat affine structures on the two-torus
• Deformations in the large of some complex manifolds, I
• Hyperkaehler manifolds with torsion, supersymmetry and Hodge theory
• Deformation types of real and complex manifolds
• Integrability of the twistor space for a hypercomplex manifold

Related

• The quaternionic/hypercomplex-correspondence
• Cohomologies on hypercomplex manifolds
• The holonomy of the Obata connection on Joyce hypercomplex manifolds
• On the type of generalized hypercomplex structures
• Complexification and hypercomplexification of manifolds with a linear connection