Mathematical Research Letters

Volume 16, Issue 2, March 2009 pp. 331-347.

Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

Authors: María L. Barberis (1), Isabel G. Dotti (2), and Misha Verbitsky (3)
Author institution: Universidad Nacional de Córdoba (1), Universidad Nacional de Córdoba (2), and Institute of Theoretical and Experimental Physics, Moscow (3)

Summary: 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 (hyperk\"ahler 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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