Geometry of Maurer-Cartan Elements on Complex Manifolds

Chen, Zhuo; Stiénon, Mathieu; Xu, Ping

doi:10.1007/s00220-010-1029-4

Cited by 10 publications

(13 citation statements)

References 34 publications

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“…It is known that T C X is a complex Lie algebroid [36]. By the Newlander-Nirenberg theorem [3], an almost complex structure J ′ is integrable if and only if gr(ξ J ′ ) = (T 0,1 X ) ′ is a Lie subalgebroid of T C X .…”

Section: Definition 427mentioning

confidence: 98%

Simultaneous deformations of a Lie algebroid and its Lie subalgebroid

2014

Journal of Geometry and Physics

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a b s t r a c tDeformation problem is an interesting problem in mathematical physics. In this paper, we show that the deformations of a Lie algebroid are governed by a differential graded Lie algebra; and under certain regularity assumptions, an L ∞ -algebra can be constructed to govern the deformations of its Lie subalgebroid. Furthermore, by applying Y. Frégier and M. Zambon's result (0000, Thm. 3), these structures can be combined together to govern the simultaneous deformations. Applications of our results include deformations of a foliation, deformations of a Lie subalgebra, deformations of a complex structure, and deformations of a homomorphism of Lie algebroids.Published by Elsevier B.V.

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Section: Definition 427mentioning

confidence: 98%

Simultaneous deformations of a Lie algebroid and its Lie subalgebroid

2014

Journal of Geometry and Physics

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“…. General results concerning such deformation theories can be found, for example, in [47,48], and a case relevant to generalized geometry has been investigated in [49].…”

Section: Sheaves Of Differential Graded Lie Algebrasmentioning

confidence: 99%

Generalized Kähler Geometry

Gualtieri

2014

Commun. Math. Phys.

104

140

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Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.A striking feature of the Courant bracket is that it has symmetries fixing the underlying space M. Any closed 2-form B ∈ Ω 2,cl (M) acts on TM, preserving the Courant bracket, via the bundle map:(1.2)Because of this symmetry, we may "twist" or modify the global structure of TM as an orthogonal bundle with a Courant bracket, keeping its local structure unchanged. These twisted structures are therefore classified by a characteristic class in H 1 (Ω 2,cl (M)) and are called exact Courant algebroids [13,18,19].

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“…Similarly, we can prove the following Theorem 3.6. Let X = CP 1 × CP 1 and π a holomorphic bivector field as in Equation (4). The Poisson cohomology of (X, π) is then given as follows.…”

Section: Poisson Cohomologymentioning

confidence: 99%