scholarly journals Hamiltonian Perspective on Generalized Complex Structure

2006 ◽  
Vol 263 (3) ◽  
pp. 711-722 ◽  
Author(s):  
Maxim Zabzine
Keyword(s):  
Complex Structure ◽  
Generalized Complex Structure ◽  
Generalized Complex

scholarly journals Generalized almost even-Clifford manifolds and their twistor spaces

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.

scholarly journals Symplectic Foliations and Generalized Complex Structures

2014 ◽  
Vol 66 (1) ◽  
pp. 31-56 ◽  
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.

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

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

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.

scholarly journals Abelian Complex Structures and Generalizations

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.

scholarly journals THREE-DIMENSIONAL TOPOLOGICAL FIELD THEORY INDUCED FROM GENERALIZED COMPLEX STRUCTURE

2007 ◽  
Vol 22 (25) ◽  
pp. 4679-4694
Author(s):  
NORIAKI IKEDA
Keyword(s):  
Complex Manifold ◽  
Mirror Symmetry ◽  
Sigma Model ◽  
Complex Structure ◽  
Three Dimensional ◽  
Poisson Manifold ◽  
Dimensional Manifold ◽  
Generalized Complex Structure ◽  
Generalized Complex ◽  
Topological Field

We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold X to an arbitrary generalized complex manifold M. The theory is invariant under the diffeomorphism on the worldvolume and the b-transformation on the generalized complex structure. Moreover the model is manifestly invariant under the mirror symmetry. We derive from this model the Zucchini's two-dimensional topological sigma model with a generalized complex structure as a boundary action on ∂X. As a special case, we obtain three-dimensional realization of a WZ-Poisson manifold.

scholarly journals GENERALIZED COMPLEX GEOMETRY AND THE POISSON SIGMA MODEL

2005 ◽  
Vol 20 (13) ◽  
pp. 985-995 ◽  
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.

scholarly journals The Local Structure of Generalized Contact Bundles

2019 ◽  
Vol 2020 (20) ◽  
pp. 6871-6925 ◽  
Author(s):  
Jonas Schnitzer ◽  
Luca Vitagliano
Keyword(s):  
Line Bundle ◽  
Symplectic Manifold ◽  
Complex Manifold ◽  
Local Structure ◽  
Complex Structure ◽  
Regular Point ◽  
Poisson Geometry ◽  
Splitting Theorem ◽  
Complex Manifolds ◽  
Generalized Complex

Abstract Generalized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

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