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Abstract: A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discretesubgroup. A complex nilmanifold is one which is equipped with a $G$-invariantcomplex structure. We prove that a complex nilmanifold has trivial canonicalbundle. This is used to study hypercomplex nilmanifolds nilmanifolds with atriple of $G$-invariant complex structures which satisfy quaternionicrelations. We prove that a hypercomplex nilmanifold admits an HKT hyperkahlerwith torsion metric if and only if the underlying hypercomplex structure isabelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balancedwith respect to all associated complex structures.



Author: Maria Laura Barberis, Isabel G. Dotti, Misha Verbitsky

Source: https://arxiv.org/







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