Holomorphic Poisson Cohomology

Zhuo Chen 1 , Daniele Grandini 2  and Yat-Sun Poon 3
  • 1 Department of Mathematical Sciences, Tsinghua University, Beijing, P.R.C.
  • 2 Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284, U.S.A
  • 3 Department of Mathematics, University of California at Riverside, Riverside, CA 92521, U.S.A

Abstract

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.

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

  • [2] C. Bartocci & E Marci, Classification of Poisson surfaces, Commun. Contemp. Math. 7 (2005) 89–95.

  • [3] S. Console, Dolbeault cohomology and deformations of nilmanifolds, Rev. de al UMA. 47 (1) (2006), 51–60.

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

  • [5] S. Console, A. Fino, & Y. S. Poon, Stability of abelian complex structures, International J. Math. 17 (2006), 401–416.

  • [6] L. A. Cordero, M. Fernández, A. Gray & L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc., 352 (2000), 5405–5433.

  • [7] D. Fiorenza & M. Manetti, Formality of Koszul brackets and deformations of holomorphic Poisson manifolds, preprint, arXiv:1109.4309v2.

  • [8] P. Gauduchon, Hermitian connections and Dirac operators, Bollettino U.M.I. 11B (1997), 257–288.

  • [9] R.Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), 525–560.

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

  • [11] G. Grantcharov, C. McLaughlin, H. Pedersen, & Y. S. Poon, Deformations of Kodaira manifolds, Glasgow Math. J. 46 (2004), 259–281.

  • [12] M. Gualtieri, Generalized complex geometry, Ann. of Math. 174 (2011), 75–123.

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

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

  • [15] N. J. Hitchin, Deformations of holomorphic Poisson manifolds, Mosc. Math. J. 669 (2012), 567–591.

  • [16] T. Höfer, Remarks on principal torus bundles, J. Math. Kyoto U., 33 (1993), 227–259.

  • [17] W. Hong, & P. Xu, Poisson cohomology of Del Pezzo surfaces, J. Algebra 336 (2011), 378–390.

  • [18] Z. J. Liu, A. Weinstein, & P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. (1997), 547–574.

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

  • [20] 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.

  • [21] K. Nomizu, On the cohomology of compact homogenous spaces of nilpotent Lie groups, Ann. Math. 59 (1954), 531–538.

  • [22] A. Polishchuk, Algebraic geometry of Poisson brackets, J. Math. Sci. (New York) 84 (1997), 1413–1444.

  • [23] Y. S. Poon, Extended deformation of Kodaira surfaces, J. reine angew. Math. 590 (2006), 45–65.

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

  • [25] S. Rollenske, Lie algebra Dolbeault cohomology and small deformations of nilmanifolds, J. London.Math. Soc. (2) 79 (2009), 346–362.

  • [26] Y. Sakane, On compact complex parallelisable solvmanifolds, Osaka J. Math. 13 (1976), 187–212.

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

  • [28] C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge studies in advanced mathematics 76 (2004), Cambridge University Press.

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