2017/02/23 by Yat Sun Poon, John Simanyi · 6 citations
Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Cohomology #Geometry and complex manifolds #Holomorphic function #Homotopy and Cohomology in Algebraic Topology #Poisson distribution #Poisson manifold #Sequence (biology) #Sheaf #Spectral sequence #math.DG #msc:14D07 #msc:18G40 #msc:32G20 #msc:53D17 #msc:53D18
paper · pdf · doi:10.1515/coma-2017-0009
published in Complex Manifolds 4(1), 137-154 (De Gruyter Open) · arXiv admin note: text overlap with arXiv:1611.08637
openalex publication_date 2017/02/23 · openalex created_date 2017/08/17 · arxiv created 2017/10/27 · arxiv updated 2017/10/31 · openalex updated_date 2026/08/05
Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-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.