Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

Abstract

We classify all the 6-dimensional unimodular Lie algebras g admitting a complex structure with non-zero closed (3,0)-form. This gives rise to 6-dimensional compact homogeneous spaces M=Γ﹨G, where Γ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-Kähler homogeneous space M=Γ﹨G admits an invariant solution with respect to some non-flat connection ∇ in the family ∇ε,ρ, then M is a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.