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Complex Manifolds

Ed. by Fino, Anna Maria

Covered by Web of Science - Emerging Sources Citation Index and Zentralblatt Math (zbMATH)

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A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Yat Sun Poon / John Simanyi
Published Online: 2017-08-04 | DOI: https://doi.org/10.1515/coma-2017-0009


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.

Keywords: Holomorphic Poisson; Cohomology; Hodge theory; Nilmanifolds

MSC 2010: 53D18; 53D17; 32G20; 18G40; 14D07


  • [1] M. Bailey, Local classification of generalize complex structures, J. Differential Geom. 95 (2013), 1-37.Google Scholar

  • [2] W. Barth, C. Peters & A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (1984) Berlin.Google Scholar

  • [3] M. L. Barberis, I. G. Dotti Miatello & R. J. Matello, On certain locally homogeneous Clifford manifolds, Ann. Glob. Anal. Geom. 13 (1995) 289-301.CrossrefGoogle Scholar

  • [4] Z. Chen, A. Fino & Y. S. Poon, Holomorphic Poisson structures and its cohomology on nilmanifolds, Differential Geom. Appl., 44 (2016), 144-160.CrossrefGoogle Scholar

  • [5] Z. Chen, D. Grandini & Y. S. Poon, Cohomology of holomorphic Poisson structures, Complex Manifolds, 2 (2015), 34-52.Google Scholar

  • [6] S. Console & A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups, 6 (2001), 111-124.Google Scholar

  • [7] P. Gauduchon, Hermitian connections and Dirac operators, Bollettino U.M.I. 11B (1997), 257-288.Google Scholar

  • [8] R. Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), 525-560.Google Scholar

  • [9] R. Goto, Unobstucted deformations of generalized complex structures induced by C1 logarithmic symplectic structures and logarithmic Poisson structures, in Geometry and Topology of manifolds, Springer Proc. Math. Stat., 154 (2016), 159-183. Preprint in arXiv:1501.03398v1.Google Scholar

  • [10] D. Grandini, Y. S. Poon & B. Rolle, Differential Gerstenhaber algebras of generalized complex structures, Asian J. Math. 18 (2014), 191-218.Google Scholar

  • [11] M. Gualtieri, Generalized complex geometry, Ann. of Math. 174 (2011), 75-123.Web of ScienceGoogle Scholar

  • [12] N. J. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003), 281-308.Google Scholar

  • [13] N. J. Hitchin, Instantons, Poisson structures, and generalized Kähler geometry, Commun. Math. Phys. 265 (2006), 131-164.Google Scholar

  • [14] N. J. Hitchin, Deformations of holomorphic Poisson manifolds, Mosc. Math. J. 669 (2012), 567-591.Google Scholar

  • [15] W. Hong, Poisson cohomology of holomorphic toric Poisson manifolds, preprint (2016) arXiv:1611.08485.Google Scholar

  • [16] W. Hong & P. Xu, Poisson cohomology of Del Pezzo surfaces, J. Algebra 336 (2011), 378-390.Web of ScienceGoogle Scholar

  • [17] C. Laurent-Gengoux, M. Stiéson & P. Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN, (2008). doi:CrossrefGoogle Scholar

  • [18] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom. 12 (1977), 253-300.Google Scholar

  • [19] Z. J. Liu, A. Weinstein & P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547-574.Google Scholar

  • [20] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Notes Series 213, Cambridge U Press, 2005.Google Scholar

  • [21] C. Maclaughlin, H. Pedersen, Y. S. Poon & S. Salamon, Deformation of 2-step nilmanifolds with abelian complex structures, J. London Math. Soc. 73 (2006) 173-193.Google Scholar

  • [22] Y. S. Poon, Extended deformation of Kodaira surfaces, J. reine angew. Math. 590 (2006), 45-65.Google Scholar

  • [23] B. Rolle, Construction of weak mirrir pairs by deformations, Ph.D. Thesis, University of California at Riverside. (2011).Google Scholar

  • [24] S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.Google Scholar

  • [25] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118 (1994) Birhäuser. ISBN 3-7643-5016-4.Google Scholar

About the article

Received: 2017-06-16

Accepted: 2017-07-12

Published Online: 2017-08-04

Published in Print: 2017-02-23

Citation Information: Complex Manifolds, Volume 4, Issue 1, Pages 137–154, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2017-0009.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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