A differential complex for Poisson manifolds

Brylinski, Jean-Luc

doi:10.4310/jdg/1214442161

Cited by 261 publications

(377 citation statements)

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“…Brylinski [3] proved that on a closed Kähler manifold (M, ω), every de Rham cohomology class has a representative α such that dα = 0, δα = 0. This implies the following…”

Section: Theorem 4 For a Poisson Manifold With Odd Betti Numbers Allmentioning

confidence: 99%

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On quantum de Rham cohomology theory

Cao

Zhou

1999

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We define the quantum exterior product ∧ h and quantum exterior differential d h on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of d h . We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d by d h and ∧ by ∧ h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.In this note we announce a construction of a deformation of the de Rham complex for any Poisson manifold. In the case of a closed symplectic manifold, its cohomology provides a deformation of the ring structure on the de Rham cohomology. More precisely, on any Poisson manifold, we define a quantum exterior product ∧ h of exterior forms, and quantum exterior differential d h , such that d 2 h = 0, and d h is a derivation for ∧ h . Here h is an indeterminate. We define the quantum de Rham cohomology as the cohomology of d h . Since d h is a derivation with respect to ∧ h , there is an induced quantum multiplication on the quantum de Rham cohomology.This work grew out of our attempt to find a new way to define quantum cohomology, which has recently attracted much attention (see Tian [13] for a survey on this topic, and the introduction of Li-Tian [11] for more recent development). Intuitively, quantum cohomology provides a deformation of the ring structure on the vector space which underlies the de Rham cohomology by counting pseudoholomorphic curves in symplectic manifolds or stable curves in algebraic manifolds. Unlike many cohomology theories in algebraic topology, it is not defined as the cohomology of a graded differential algebra. For the sake of being consistent with other cohomology theories, it would be desirable to be able to do so, even though the applications of quantum cohomology do not really require this property. This is the first motivation for our construction of quantum de Rham cohomology. On the other hand, from the point of view of finding deformations of the ring structure on the de Rham cohomology per se, it is also interesting to see whether deformations of the de Rham complex can provide new deformations of the de Rham cohomology. This is the second motivation to our construction.

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“…Brylinski [3] proved that on a closed Kähler manifold (M, ω), every de Rham cohomology class has a representative α such that dα = 0, δα = 0. This implies the following…”

Section: Theorem 4 For a Poisson Manifold With Odd Betti Numbers Allmentioning

confidence: 99%

“…We begin with a 2n-dimensional symplectic vector space (V, ω). Brylinski [3] defined a symplectic star operator * : Λ k (V * ) → Λ 2n−k (V * ). We can extend it to Λ h,h −1 by setting * h = h −1 , and…”

Section: Quantum Hard Lefschetz Theoremmentioning

confidence: 99%

On quantum de Rham cohomology theory

Cao

Zhou

1999

Electron. Res. Announc. Amer. Math. Soc.

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“…The Poisson homology H • (A, ∂ π ) of (A, π) is that of the complex (Γ(∧ • A * ), ∂ π ) which has been studied by Huebschmann [55], and which generalizes the Poisson homology of Poisson manifolds [80,12].…”

Section: Lie Algebroids With a Poisson Structurementioning

confidence: 99%

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Kosmann–Schwarzbach¹

2008

SIGMA

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“…For details, see [3,15,21]. The Hamiltonian G-manifolds (M, Ω) we construct here have the property that each de Rham cohomology class contains a symplectic harmonic representative while the same is not true for (M G, ω).…”

Section: Introductionmentioning

confidence: 99%