Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

BARBERIS M. L., DOTTI I.G., VERBITSKY M.

MATHEMATICAL RESEARCH LETTERS

International Press

Año: 2008

1073-2780

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compactdiscrete subgroup. A complex nilmanifold is one which is equippedwith a $G$-invariant complex structure. We prove that a complexnilmanifold has trivial canonical bundle. This is used to studyhypercomplex nilmanifolds (nilmanifolds with a triple of$G$-invariant complex structures which satisfy quaternionicrelations). We prove that a hypercomplex nilmanifold admits an HKT(hyperk"ahler with torsion) metric if and only if   the underlyinghypercomplex structure is abelian. Moreover, any $G$-invariantHKT-metric on a nilmanifold is balanced with respect to allassociated complex structures.