STABILITY OF ABELIAN COMPLEX STRUCTURES

By S. Console, Anna Fino, Yat Sun Poon. 2005

55 citations recorded by the source corpus.

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 [7] 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c64d7d1d9657d7f2742ad8249e5625b694a0b449

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Hermitian structures on six-dimensional nilmanifolds (2004, background): “Recently, the result of [MPPS] has been generalized to any nilmanifold with abelian complex structure in [ CFP ], where it is also given an example of an abelian complex structure on a tendimensional 3-step nilmanifold which deforms into nonnilpotent complex structures.”
Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry (2007, unknown): “ the Dolbeault cohomology of an abelian nilmanifold can be expressed in terms of the corresponding Lie algebra cohomology ([CF], [CFGU]), and the same is true for the deformation space ([MPPS], [V2], [CFP]). This notion is specially convenient when applied to hypercomplex nilmanifolds. If (N,I,J,K) ”
Geometry of nilmanifolds with left‐invariant complex structure and deformations in the large (2009, background): “Generalising results on deformations of abelian complex structures [8, 21] we proved the following theorem.”