Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

Abstract

A nilmanifold is a quotient of a nilpotent group G by a cocompact 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 (hyperkähler 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.

Chern-flat and Ricci-flat invariant almost Hermitian structures

“…Abelian complex structures were introduced in [5] and were intensely studied in [4,6,11,8,19]. Observe that J is anti-abelian and integrable if and only if it is bi-invariant, i.e.…”

The Kuranishi space of complex parallelisable nilmanifolds

“…This follows, as noted in [BDV07], directly from Salamon's work [Sal01,Theorem 3.1] and is also true in a slightly more general context [CG04].…”

Non-Kähler Calabi-Yau manifolds

“…A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle. The geometry of compact complex manifolds with trivial canonical bundle has been investigated for example by [2,6,12,19,21,25,26,27,30,33,45,52,62,65] and others. In this paper we will consider a more general class of manifolds, that we now define, and argue that they can naturally be considered as "non-Kähler Calabi-Yau" manifolds.…”

“…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. For example, every compact complex nilmanifold with a left-invariant complex structure has trivial canonical bundle, and it is always non-Kähler unless it is a torus [6]. A lot of interest in the subject was generated by "Reid's fantasy" [64] that all Calabi-Yau threefolds with trivial canonical bundle should form a connected family provided one allows deformations and singular transitions through non-Kähler manifolds with trivial canonical bundle.…”