Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

Details Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

2007-12-22

arxiv.org/abs/0712.3863v4

Math. Res. Lett. 16 (2009), no. 2, 331--347.

Mathematics

Differential Geometry

19 pages, v. 4, added reference to arXiv:0803.2048 by S. Rollenske

Scientific paper

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 (hyperkahler 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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