Differential Gerstenhaber algebras of generalized complex structures

Grandini, Daniele; Poon, Yat Sun; Rolle, Brian

doi:10.4310/ajm.2014.v18.n2.a1

Cited by 7 publications

(10 citation statements)

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“…The differential ∂ has the properties such that when it is restricted to (0, 1)-forms, it is the classical ∂-operator in complex manifold theory; meaning that it is the (0, 2)-component of the exterior differential [11]. Similarly, when the Lie algebroid differential is restricted to (1, 0)-vector fields, it is the Cauchy-Riemann operator as seen in [8].…”

Section: Algebraic Structures On Poisson Cohomologymentioning

confidence: 99%

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Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Poon

Simanyi

2019

Complex Manifolds

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It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

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Section: Algebraic Structures On Poisson Cohomologymentioning

confidence: 99%

“…In this vein, one studies deformation within the framework of Lie bi-algebroids [20]. There has been work along this line by various authors [9] [10] [11] [16] [17] [18] [24]. In this note, the authors continue to treat holomorphic Poisson structures within the realm of generalized complex structures, and extend the work in [24].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Poon

Simanyi

2019

Complex Manifolds

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“…Recently, Hong studied holomorphic Poisson cohomology on toric Poisson manifolds [15]. For nilmanifolds with abelian complex structures, there is recent work done by this author and his collaborators [4,5,10]. A common feature of these works is to recognize the cohomology as the hypercohomology of a bi-complex.…”

Section: Introductionmentioning

confidence: 99%

A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Poon

Simanyi

2017

Complex Manifolds

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A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical ∂-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.Since the center c is J-invariant and it contains g k−1 ,So, the filtration (18) splits into two. One is for type (1, 0)-vectorsAnother is for type (0, 1)-vectorsLemma 1 [4, Lemma 4] Suppose that the complex structure J is abelian. Then • [[g (1,0) , g (1,0) ]] = 0, and [[g (0,1) , g (0,1) ]] = 0.For the quotient space, we will adopt the notationChoose a vector space isomorphism so that the short exact sequence of Lie algebras 0 → g ℓ,(1,0) J → g ℓ−1,(1,0) J → g ℓ−1,(1,0) J /g ℓ,(1,0) J → 0 is turned into a direct sum of vector spaces. g ℓ−1,(1,0) J ∼ = t (1,0) ℓ ⊕ g ℓ,(1,0) J

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“…In recent years, there are significant interests on holomorphic Poisson structures due to their emergence in generalized complex geometry [11] [12] [13]. As such, one could consider their deformations either as complex analytic objects [6] [8] [14], or as generalized complex objects [9]. On the other hand, there are classifications of holomorphic Poisson structures on algebraic surfaces [2], and computations of their related cohomology theory [16].…”

Section: Introductionmentioning

confidence: 99%

“…A holomorphic Poisson structure consists of a holomorphic bi-vector field Λ on a complex manifold such that [[Λ, Λ]] = 0. It is demonstrated in [9] that the deformation of such structure as a generalized complex structure is dictated by the cohomology space ⊕ k H k Λ of the differential operator ∂ Λ = ∂ + [[Λ, −]]. This observation motivates the authors' desire to find a way to compute the cohomology spaces H k Λ .…”

Section: Introductionmentioning

confidence: 99%

Holomorphic Poisson Cohomology

2015

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Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault ∂-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures. IntroductionThe algebraic geometry of Poisson brackets over complex manifolds were studied for some time [22]. In recent years, there are significant interests on holomorphic Poisson structures due to their emergence in generalized complex geometry [12] . This observation motivates the authors' desire to find a way to compute the cohomology spaces H k Λ . Since this space is naturally the limit of a spectral sequence of a double complex due to the operators ∂ and [ [Λ, −]], it is an obvious question on when and how fast this spectral sequence could degenerate. We will see that the first sheet of this spectral sequence is the Dolbeault cohomology on the manifold with coefficients in the holomorphic polyvectors. As this is a classical object, it becomes desirable to identify the conditions under which the spectral sequence will degenerate at its second level.In this paper, we identify three situations in which the spectral sequence of the holomorphic Poisson double complex degenerates at its second level. In the next chapter, we will explain the role of holomorphic Poisson structures in generalized geometry, and then set up our computation of cohomology in the context of Lie bi-algebroids and their dual differentials [18]. Our reference for differential calculus of Lie algebroids is the book [19]. We will then set up the holomorphic Poisson double complex and its related spectral sequence.We will make observation on three very different kinds of complex manifolds: complex parallelizable, nilmanifolds with abelian complex structures, and Kählerian manifolds. This theorem will appear as Theorem 6. Note that it is well known that the Frölicher spectral sequence on a compact complex surface always degenerates [1]. The last theorem could be applied to all compact complex surfaces as well as all compact Kählerian manifolds [28]. Holomorphic Poisson Double Complex Holomorphic Poisson Structures as Generalized Complex StructuresA generalized complex manifold [12] [13] is a smooth 2n-dimensional manifold M equipped with a subbundle L of the bundle T = (TM ⊕ T * M) C such that -L and its conjugate bundle L are transversal; -L is maximally isotropic with...

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Differential Gerstenhaber algebras of generalized complex structures

Cited by 7 publications

References 24 publications

Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Holomorphic Poisson Cohomology

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