Hypercomplex Structures on a Class of Solvable Lie Groups

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.

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“…Consequently the integrability condition for a deformation Φ is, for any ω ∈ g * (1,0) andX,Ȳ ∈ g 0,1 ,…”

Section: Deformation Theorymentioning

confidence: 99%

Stability of Abelian Complex Structures

2006

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“…In the above computation, we repeatedly use the commutative law (1), distributive laws (2) and (3), and information on degree-one elements to generate all the algebraic relations among elements of higher degrees. It will be useful to summarize all the 'generating data' for the algebra.…”

Section: Dga Of Kodaira Surfacesmentioning

confidence: 99%

Extended deformation of Kodaira surfaces

Poon

2006

Journal Fur Die Reine Und Angewandte Mathematik (Crelles Journal)

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We present the extended Kuranishi space for Kodaira surface as a nontrivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image in the sense of Merkulov. The calculations of extended deformation and the weak Frobenius structure are based on Merkulov's perturbation method. Our computation of cohomology is done in the context of compact nilmanifolds.

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“…A number of authors have studied Lie groups as manifolds equipped with various tensor structures and metrics that are compatible with the structures (including in the lowest-dimensional cases) -for example, [3] and [4] for almost contact metric manifolds, [13] and [10] for almost contact B-metric manifolds, [1] and [6] for almost complex manifolds with Hermitian metric, [8] and [24] for almost complex manifolds with Norden metric, [2] and [5] for hypercomplex hyper-Hermitian manifolds, [7] and [12] for almost hypercomplex Hermitian-Norden manifolds, [9] and [22] for Riemannian almost product manifolds, [16] and [25] for almost paracontact metric manifolds.…”

Section: Introductionmentioning

confidence: 99%