Extended deformation of Kodaira surfaces

Poon, Yat Sun

doi:10.1515/crelle.2006.003

Cited by 21 publications

(56 citation statements)

References 23 publications

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“…If we use the notation x to denote (x, 0) in h, then the above identity becomes −Jγ (x)Jy = γ (x)y. Therefore, one may consider the representation γ playing both the role of representation ρ and the role of the connection γ in (8). This coincidence is the consequence of V being the underlying vector space of the Lie algebra g.…”

Section: Definition 42mentioning

confidence: 99%

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Weak mirror symmetry of complex symplectic Lie algebras

Cleyton

Poon

Ovando

2011

Journal of Geometry and Physics

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MSC:primary 53D37 secondary 14J33 53D05 53D12 32G81 Keywords:Weak mirror symmetry Complex symplectic algebras Gerstenhaber algebras Symplectic connections Flat connections a b s t r a c t A complex symplectic structure on a Lie algebra h is an integrable complex structure J with a closed non-degenerate (2, 0)-form. It is determined by J and the real part Ω of the (2, 0)-form. Suppose that h is a semi-direct product g V , and both g and V are Lagrangian with respect to Ω and totally real with respect to J. This note shows that g V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω and J are isomorphic.The geometry of (Ω, J) on the semi-direct product g V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra g. By further exploring a relation between (J, Ω) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8n from the data of the 4n-dimensional ones.

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Section: Definition 42mentioning

confidence: 99%

“…However, the work in [6] does not explain an elementary, but non-trivial example on the Kodaira-Thurston surface, a real four-dimensional example [8]. Yet the Kodaira-Thurston surface is a key example of hypersymplectic structures [9,10].…”

Section: Introductionmentioning

confidence: 99%

Weak mirror symmetry of complex symplectic Lie algebras

Cleyton

Poon

Ovando

2011

Journal of Geometry and Physics

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“…If one extends the Schouten-Nijenhuis bracket {·, ·} to the exterior algebra by antiderivative as seen in [7] and make use of the assumption that J is abelian, a short computation shows that…”

Section: Deformation Theorymentioning

confidence: 99%

Stability of Abelian Complex Structures

2006

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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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“…A feature of the theory of generalized complex structures is a deformation theory allowing extrapolation between complex and symplectic structures [9] [17]. It is in sharp contrast to the well known fact that deformation of symplectic structures on compact manifolds are trivial [16].…”

Section: Discussionmentioning

confidence: 99%