Stability of Abelian Complex Structures

Details Stability of Abelian Complex Structures Stability of Abelian Complex Structures

2005-06-15

arxiv.org/abs/math/0506314v1

Mathematics

Differential Geometry

17 pages

Scientific paper

Let $M = \Gamma \backslash G$ 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 of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon 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.

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