A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Abstract

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

Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

“…It also extends and explains various examples of two-steps nilmanifolds found in [24]. At the end of this article we provide abundant examples beyond two-step nilmanifolds on which the conditions for Theorem 5 are satisfied.…”

“…Remark The above observation corrects an error in the statement of Proposition 9.2 in [24] for omitting a requirement on the step of the nilmanifold.…”

“…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].…”

Frobenius Structures and Generalized Deformation of Kodaira Manifolds

“…Based on such foundation, the author and his collaborators recently identify when the spectral sequence degenerates on the first page if the complex structure is abelian and the underlying nilmanifold is 2-steps [48] [49]. It enables an effective computation of holomorphic Poisson cohomology.…”

Holomorphic Koszul-Brylinski homologies of Poisson blow-ups

“…where the differential operator b π (−) = [π, −] is the adjoint action of π with respect to the Schouten bracket; see [25,23]. This cohomology has been widely studied; see, for example, [20,12,10,7,27,28,19] and references therein. Dually, from a homological point of view, we have the so called holomorphic Koszul-Brylinski complex:…”