Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle

Abstract

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family (M, Ja) of compact complex manifolds such that (M, Ja) satisfies the ∂∂lemma and admits a balanced metric for any a = 0, but the central limit neither satisfies the ∂∂-lemma nor admits balanced metrics.

Cohomological decomposition of compact complex manifolds and holomorphic deformations

“…We will use the results about 6-dimensional solvmanifolds with holomorphically trivial canonical bundle obtained in [15]. Concretely, let G be the 6-dimensional simply-connected solvable Lie group whose Lie algebra g is defined by the following structure equations: In this section we use the classification of complex nilmanifolds satisfying the C ∞ -pure-and-full property obtained in Section 2 to illustrate that this property is unrelated to other metric properties.…”

“…Based on a result in [15], in the following theorem we show that there is a lattice and a holomorphic deformation of the complex structure J showing that being C ∞ -pure-and-full is not a closed property.…”

“…Furthermore, the central limit of an analytic family of compact balanced manifolds may not have any strongly Gauduchon metric. In [15] (see also [4]) one can find an analytic family of compact balanced ∂∂-manifolds X t such that the central limit X 0 neither satisfies the ∂∂-Lemma nor admits balanced metrics. Here we show that the C ∞ -pure-andfull property is not deformation closed.…”

“…In Section 4 we prove that the C ∞ -pure-and-full property is not closed under holomorphic deformations. Using the study of 6-dimensional solvmanifolds with holomorphically trivial canonical bundle [15], we find an analytic family of compact complex manifolds X t such that X t is C ∞ -pure-and-full for every t = 0, but X 0 is neither C ∞ -pure nor C ∞ -full (see Theorem 4.1). Therefore, for compact complex manifolds, the properties of being C ∞ -pure-and-full, being C ∞ -pure, and being C ∞ -full are not closed under holomorphic deformations.…”

Extending invariant complex structures

“…This is an open topic of active research now, as for instance the six-dimensional situation starting by the existence problem of such structures (see [2,18,20,21,24] for advances in this direction). Indeed the existence and the classification problems become more complicated in higher dimensions.…”