Deformation of 2-Step Nilmanifolds with Abelian Complex Structures
نویسندگان
چکیده
We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in 3 complex dimensions. AMS Subject Classification: 32G05; 53C15, 53C56, 57S25, 17B30
منابع مشابه
Stability of Abelian Complex Structures
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 structu...
متن کاملHyperkähler Torsion Structures Invariant by Nilpotent Lie Groups
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex struc...
متن کاملClassification of abelian complex structures on 6-dimensional Lie algebras
Let g be a Lie algebra, J an endomorphism of g such that J = −I , and let g be the ieigenspace of J in g := g ⊗R C. When g is a complex subalgebra we say that J is integrable, when g is abelian we say that J is abelian and when g is a complex ideal we say that J is bi-invariant. We note that a complex structure on a Lie algebra cannot be both abelian and biinvariant, unless the Lie bracket is t...
متن کاملOn 5-dimensional 2-step homogeneous randers nilmanifolds of Douglas type
In this paper we first obtain the non-Riemannian Randers metrics of Douglas type on two-step homogeneous nilmanifolds of dimension five. Then we explicitly give the flag curvature formulae and the $S$-curvature formulae for the Randers metrics of Douglas type on these spaces. Moreover, we prove that the only simply connected five-dimensional two-step homogeneous Randers nilmanifolds of D...
متن کامل2 6 Fe b 20 04 Kodaira Surface , its Extended Deformation and Mirror Image
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. Our computation is done in the context of compact nilmanifolds. The calculation of extended deformation and ...
متن کامل