Stability of Abelian Complex Structures

Abstract

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [5] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

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

“…We show in Section 5 that this is the case if and only if they are infinitesimally complex parallelisable; the analogous results holds for abelian complex structures [CFP06].…”

“…If we need to access the underlying real object with left-invariant complex structure we will write for example g = (h, J ). By the above lemma we can then identify for all x, y ∈ h. In some sense this is the opposite condition to being a complex Lie algebra and their deformations have been studied in [MPPS06,CFP06]. As we pointed out in the introduction, deformations behave much more nicely in this case.…”

“…tangent sheaf) (M,J ) is of particular interest. It has been calculated in [Rol08] for left-invariant complex structures for which (4) is an isomorphism, generalising results on abelian complex structure in [MPPS06,CFP06].…”

“…Many known examples like complex tori, the Iwasawa manifold [Nak75], Kodaira surfaces [Bor84], abelian complex structures [CFP06,MPPS06] (see Section 2 for a definition) have indeed unobstructed deformations, and the speculation that a kind of TianTodorov Lemma could be true for large classes of nilmanifolds, including for example hypercomplex structures, was supported by results on weak homological mirror symmetry for nilmanifolds [Poo06,CP08]. On the other hand Catanese and Frediani observed in their study of deformations of principal holomorphic torus bundles, which are in particular nilmanifolds with left-invariant complex structure, that the Kuranishi space can be singular [CF06].…”