Half-flat nilmanifolds

We introduce a double complex that can be associated to certain Lie algebras, and show that its cohomology determines an obstruction to the existence of a half-flat SU(3)-structure. We obtain a classification of the 6-dimensional nilmanifolds carrying an invariant half-flat structure. MSC classification: Primary 53C25; Secondary 53C29, 17B30 An SU(3)-structure on a manifold of real dimension 6 consists of a Hermitian structure (g, J, ω) and a unit (3, 0)-form Ψ; since SU(3) is the stabilizer of the transitive action of G2 on S , it follows that a G2-structure on a 7-manifold induces an SU(3)-structure on any oriented hypersurface. If the G2-structure is torsion-free, meaning that it corresponds to a holonomy reduction, then the SU(3)-structure is half-flat [3]; in terms of the defining forms, this means dω ∧ ω = 0, dReΨ = 0. (1) Conversely, it follows from a result of Hitchin [10] that every compact, realanalytic half-flat 6-manifold can be realized as a hypersurface in a manifold with holonomy contained in G2, though this is no longer true if the real-analytic hypothesis is dropped [1]. Moreover, the G2-structure can be obtained from the half-flat structure by solving an ODE, so that the construction of half-flat structures is indirectly a means of constructing local metrics with holonomy G2. Half-flat manifolds are also studied in string theory (see e.g. [8]). An effective technique to obtain compact examples of half-flat manifolds consists in considering left-invariant structures on a nilpotent Lie group, following [13]. Six-dimensional nilpotent Lie algebras are classified in [11]; moreover, each associated Lie group has a uniform discrete subgroup (see [12]), giving rise to a compact quotient, called a nilmanifold. Thus, an SU(3)-structure on the Lie algebra determines an invariant SU(3)-structure on the associated nilmanifold, and vice versa. The nilmanifolds admitting certain special types of half-flat structures have been classified in [4, 2, 6], and an analogous classification in five dimensions has been obtained in [5]; however, the problem of determining all the nilmanifolds admitting an invariant half-flat structure has been open for some time. A related 5-dimensional geometry has been studied in [7], leading to examples of half-flat structures on solvable Lie algebras.