Mathematical Research Letters

Volume 16 (2009)

Number 2

Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

Pages: 331 – 347



María L. Barberis (Universidad Nacional de Córdoba)

Isabel G. Dotti (Universidad Nacional de Córdoba)

Misha Verbitsky (Institute of Theoretical and Experimental Physics, Moscow)


A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkähler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.

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