VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

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Symplectic Foliations and Generalized Complex Structures

Canadian Journal of Mathematics ◽

10.4153/cjm-2013-007-6 ◽

2014 ◽

Vol 66 (1) ◽

pp. 31-56 ◽

Cited By ~ 4

Author(s):

Michael Bailey

Keyword(s):

Symplectic Form ◽

Fibre Bundle ◽

Complex Structure ◽

Sufficient Conditions ◽

Affirmative Answer ◽

Generalized Complex Structures ◽

Generalized Complex Structure ◽

Generalized Complex ◽

Necessary And Sufficient ◽

Obstruction Class

AbstractWe answer the natural question: when is a transversely holomorphic symplectic foliation induced by a generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base that does not come from a generalized complex structure, and a regular generalized complex structure that is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.

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AV-Courant Algebroids and Generalized CR Structures

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-009-1 ◽

2011 ◽

Vol 63 (4) ◽

pp. 938-960 ◽

Cited By ~ 9

Author(s):

David Li-Bland

Keyword(s):

Contact Structure ◽

Complex Structures ◽

Lie Algebroid ◽

Courant Algebroids ◽

The Third ◽

Cr Structure ◽

Generalized Complex Structures ◽

Generalized Complex ◽

Cr Structures

Abstract We construct a generalization of Courant algebroids that are classified by the third coho- mology group H3(A , V), where A is a Lie Algebroid, and V is an A-module. We see that both Courant algebroids and structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure.

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Reduction of Courant algebroids and generalized complex structures

Advances in Mathematics ◽

10.1016/j.aim.2006.09.008 ◽

2007 ◽

Vol 211 (2) ◽

pp. 726-765 ◽

Cited By ~ 88

Author(s):

Henrique Bursztyn ◽

Gil R. Cavalcanti ◽

Marco Gualtieri

Keyword(s):

Complex Structures ◽

Courant Algebroids ◽

Generalized Complex Structures ◽

Generalized Complex

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Abelian Complex Structures and Generalizations

Complex Manifolds ◽

10.1515/coma-2020-0117 ◽

2021 ◽

Vol 8 (1) ◽

pp. 247-266

Author(s):

Yat Sun Poon

Keyword(s):

Deformation Theory ◽

Complex Structure ◽

Complex Structures ◽

Generalized Complex Structure ◽

Generalized Complex

Abstract After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.

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GENERALIZED COMPLEX GEOMETRY AND THE POISSON SIGMA MODEL

Modern Physics Letters A ◽

10.1142/s0217732305017184 ◽

2005 ◽

Vol 20 (13) ◽

pp. 985-995 ◽

Cited By ~ 13

Author(s):

L. BERGAMIN

Keyword(s):

Sigma Model ◽

Complex Geometry ◽

Complex Structure ◽

Complex Structures ◽

Almost Complex Structure ◽

Poisson Structures ◽

Generalized Complex Structures ◽

Almost Complex ◽

Generalized Complex ◽

Poisson Sigma Model

The supersymmetric Poisson Sigma model is studied as a possible worldsheet realization of generalized complex geometry. Generalized complex structures alone do not guarantee non-manifest N = (2, 1) or N = (2, 2) supersymmetry, but a certain relation among the different Poisson structures is needed. Moreover, important relations of an additional almost complex structure are found, which have no immediate interpretation in terms of generalized complex structures.

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Generalized almost even-Clifford manifolds and their twistor spaces

Complex Manifolds ◽

10.1515/coma-2020-0108 ◽

2021 ◽

Vol 8 (1) ◽

pp. 96-124

Author(s):

Luis Fernando Hernández-Moguel ◽

Rafael Herrera

Keyword(s):

Twistor Space ◽

Complex Structure ◽

Twistor Spaces ◽

Recent Interest ◽

Generalized Complex Structure ◽

Generalized Complex ◽

Space Construction ◽

Clifford Structures

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

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Complex structures on the complexification of a real Lie algebra

Complex Manifolds ◽

10.1515/coma-2018-0010 ◽

2018 ◽

Vol 5 (1) ◽

pp. 150-157

Author(s):

Takumi Yamada

Keyword(s):

Lie Algebra ◽

Complex Structure ◽

Complex Structures ◽

Direct Sum ◽

Direct Sum Decomposition

AbstractLet g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ(gℂ) from the decomposition, where ℝ(gℂ) is a real Lie algebra obtained from gℂby the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ(gℂ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim Hs.t∂̄J̃ (hℂ).

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