Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access August 4, 2017

A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Yat Sun Poon EMAIL logo and John Simanyi
From the journal Complex Manifolds


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.

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


[1] M. Bailey, Local classification of generalize complex structures, J. Differential Geom. 95 (2013), 1-37.10.4310/jdg/1375124607Search in Google Scholar

[2] W. Barth, C. Peters & A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (1984) Berlin.10.1007/978-3-642-96754-2Search in 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) 289-301.10.1007/BF00773661Search in Google Scholar

[4] Z. Chen, A. Fino & Y. S. Poon, Holomorphic Poisson structures and its cohomology on nilmanifolds, Differential Geom. Appl., 44 (2016), 144-160.10.1016/j.difgeo.2015.11.006Search in Google Scholar

[5] Z. Chen, D. Grandini & Y. S. Poon, Cohomology of holomorphic Poisson structures, Complex Manifolds, 2 (2015), 34-52.10.1515/coma-2015-0005Search in Google Scholar

[6] S. Console & A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups, 6 (2001), 111-124.10.1007/BF01597131Search in Google Scholar

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

[8] R. Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), 525-560.10.4310/jdg/1279114300Search in 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.Search in Google Scholar

[10] D. Grandini, Y. S. Poon & B. Rolle, Differential Gerstenhaber algebras of generalized complex structures, Asian J. Math. 18 (2014), 191-218.10.4310/AJM.2014.v18.n2.a1Search in Google Scholar

[11] M. Gualtieri, Generalized complex geometry, Ann. of Math. 174 (2011), 75-123.10.4007/annals.2011.174.1.3Search in Google Scholar

[12] N. J. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003), 281-308.10.1093/qmath/hag025Search in Google Scholar

[13] N. J. Hitchin, Instantons, Poisson structures, and generalized Kähler geometry, Commun. Math. Phys. 265 (2006), 131-164.10.1007/s00220-006-1530-ySearch in Google Scholar

[14] N. J. Hitchin, Deformations of holomorphic Poisson manifolds, Mosc. Math. J. 669 (2012), 567-591.10.17323/1609-4514-2012-12-3-567-591Search in Google Scholar

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

[16] W. Hong & P. Xu, Poisson cohomology of Del Pezzo surfaces, J. Algebra 336 (2011), 378-390.10.1016/j.jalgebra.2010.12.017Search in Google Scholar

[17] C. Laurent-Gengoux, M. Stiéson & P. Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN, (2008). doi: 10.1093/imrn/rnn088Search in Google Scholar

[18] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom. 12 (1977), 253-300.10.4310/jdg/1214433987Search in Google Scholar

[19] Z. J. Liu, A. Weinstein & P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547-574.10.4310/jdg/1214459842Search in 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, 2005.10.1017/CBO9781107325883Search in 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) 173-193.10.1112/S0024610705022519Search in Google Scholar

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

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

[24] S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.10.1016/S0022-4049(00)00033-5Search in Google Scholar

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

Received: 2017-6-16
Accepted: 2017-7-12
Published Online: 2017-8-4
Published in Print: 2017-2-23

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 1.2.2023 from
Scroll Up Arrow