Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

• Autores: María L. Barberis, Isabel G. Dotti, Misha Verbitsky
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 16, Nº 2-3, 2009, págs. 331-347
• Idioma: inglés
• DOI: 10.4310/mrl.2009.v16.n2.a10
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• Resumen

  • 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.