Stability of Abelian Complex Structures

2005/06/15 by Sergio Console, Console, Sergio, Anna Fino +3 · 2 citations
Mathematics · #22E25 #32G05 #53C15 #53C56 #57S25 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.DG #msc:22E25 #msc:32G05 #msc:53C15 #msc:53C56 #msc:57S25

paper · pdf · doi:10.48550/arxiv.math/0506314

17 pages

arxiv created 2005/06/15 · arxiv updated 2009/12/01

Abstract

Let M = Γ\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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• On splittings of deformations of pairs of complex structures and holomorphic vector bundles
• Complex structures on two-step nilpotent Lie groups

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