Generalized deformation of complex structures on nilmanifolds

Abstract

On a 2-step nilmanifold with abelian complex structure, there exists an invariant (1,0)-form ρ such that dρ is type-(1,1). It acts on the kernel of ρ by contraction. When this contraction map is non-degenerate, for any given infinitesimal generalized complex deformation Γ1 we construct a solution (Γ,∂→) for the extended Maurer-Cartan equation. It amounts to identifying the obstruction ∂→ for Γ1 to be integrable, and constructing the deformation Γ when the obstruction vanishes. As a consequence of our explicit solutions for (Γ,∂→), we prove that on any real six-dimensional 2-step nilmanifold with abelian complex structure, when the contraction map is non-degenerate, every infinitesimal generalized complex deformation sufficiently close to zero is integrable. We also show that in all dimensions, if the contraction map is skew-Hermitian, then every infinitesimal generalized complex deformation sufficiently close to zero is integrable. Moreover the differential graded algebra controlling the generalized deformation of the underlying abelian complex structure is quasi-isomorphic to the one after deformation.