Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

By M. L. Barberis, I. Dotti, Misha Verbitsky. 2007

79 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1ab37ef0152b9071415c0ef9059b680e93ba0c10

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In Applied Mathematics

1. R. Villacampa
2. A. Otal
3. Manuel Ceballos
4. Valentino Tosatti
5. L. Ugarte
6. Xiaokui Yang
7. B. Weinkove
8. Yuguang Zhang

In Geometry and Topology

1. R. Villacampa
2. A. Otal
3. Manuel Ceballos
4. Valentino Tosatti
5. L. Ugarte
6. Xiaokui Yang
7. B. Weinkove
8. Yuguang Zhang

In Mathematical Physics

1. Valentino Tosatti
2. R. Villacampa
3. Manuel Ceballos
4. L. Ugarte

Citations in the corpus

• Non-Kähler Calabi-Yau manifolds (2014, background): “On the other hand, much interest has been devoted to studying non-Kähler compact complex manifolds with holomorphically trivial (or more generally torsion) canonical bundle, and many examples can be found in [6, 15, 17, 20, 23, 33, 35, 36, 37, 40, 43, 53, 60, 78, 83, 86] and references therein.”
• Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics (2011, background): “try have been investigated, as for instance the Dolbeault cohomology [7, 13, 26], complex deformations [6, 8, 20, 27] or the existence of special Hermitian metrics [16, 30]. Recently, it is proved in [4] that the canonical 1 2 bundle of any complex nilmanifold is holomorphically trivial and some app”