Journal of the Mathematical Society of Japan

Poisson structures and generalized Kähler submanifolds

Ryushi GOTO

Full-text: Open access

Abstract

Let $X$ be a compact Kähler manifolds with a non-trivial holomorphic Poisson structure $\beta$ . Then there exist deformations $\{(\mathscr{J}_{\beta t}, \psi_t)\}$ of non-trivial generalized Kähler structures with one pure spinor on $X$ . We prove that every Poisson submanifold of $X$ is a generalized Kähler submanifold with respect to $(\mathscr{J}_{\beta t}, \psi_t)$ and provide non-trivial examples of generalized Kähler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bihermitian structures constructed from Poisson structures.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 1 (2009), 107-132.

Dates
First available in Project Euclid: 9 February 2009

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1234189030

Digital Object Identifier
doi:10.2969/jmsj/06110107

Mathematical Reviews number (MathSciNet)
MR2272873

Subjects
Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 32J27: Compact Kähler manifolds: generalizations, classification 53D17: Poisson manifolds; Poisson groupoids and algebroids

Keywords
generalized complex generalized Kähler structures and Poisson structures

Citation

GOTO, Ryushi. Poisson structures and generalized Kähler submanifolds. Journal of the Mathematical Society of Japan 61 (2009), no. 1, 107--132. doi:10.2969/jmsj/06110107. http://projecteuclid.org/euclid.jmsj/1234189030.


Export citation

References

  • V. Apostolov, P. Gauduchon and G. Grantcharov, Bihermitian structures on complex surfaces, Proc. London Math. Soc., 79 (1999), 414–428, Corrigendum: 92 (2006), 200–202.
  • S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Int. Math. Res. Not., (1998), 201–215.
  • J. Barton and M. Stiénon, Generalized complex submanifols, Math.DG/0603480
  • O. Ben-Bassart and M. Boyarchenko, Submanifolds of generalized complex manifolds, J. Symplectic Geom., 2 (2004), 309–355.
  • H. Bursztyn, M. Gualtieri and G. Cavalcanti, Reduction of Courant algebroids and generalized complex structures, Adv. Math., 211 (2007), 726–765, Math.DG/0509640
  • G. Cavalcanti, New aspects of $dd^c$-lemma, Math.DG/0501406
  • C. C. Chevalley, The algebraic theory of Spinors, Columbia University Press, 1954.
  • T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631–661.
  • J. P. Dufour and N. T. Zung, Poisson structures and their normal forms, Progress in Math., 242, Birkhäuser, 2000.
  • A. Fujiki and M. Pontecorvo, Bihermitian anti-self-dual structures on Inoue surfaces, preprint, 2007.
  • R. Goto, Moduli spaces of topological calibrations, Calabi-Yau, hyperKähler, G$_2$ and Spin$(7)$ structures, Interna. J. Math., 115 (2004), 211–257.
  • R. Goto, On deformations of generalized Calabi-Yau, hyperKähler, G$_2$ and Spin$(7)$ structures, Math.DG/0512211
  • R. Goto, Deformations of \complex and \Kähler structures, Math. DG/0705.2495
  • M. Gualtieri, Generalized complex geometry, Math.DG/0703298
  • M. Gualtieri, Hodge decomposition for generalized Kähler manifolds, Math. DG/0409093
  • M. Gualtieri, Branes and Poisson varieties, Math.DG/0710.2719
  • N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., 54 (2003), 281–308, Math. DG/0401221
  • N. Hitchin, Instantons, Poisson structures and generalized Kähler geometry, Commun. Math. Phys., 265 (2006), 131–164.
  • N. Hitchin, Bihermitian metrics on Del Pezzo surfaces, Math.DG/060821
  • D. Huybrechts, Generalized Calabi-Yau structures, $K3$ surfaces and B-fields, math.AG/0306132, Interna. J. Math., 16 (2005), 13–36.
  • K.Kodaira, Complex manifolds and deformations of complex structures, Grundlehren der Mathematischen Wissenschaften, 283, springer-Verlag, 1986.
  • K. Kodaira and D. C. Spencer, On deformations of complex, analytic structures I, II, Ann. of Math., 67 (1958), 328–466.
  • K. Kodaira and D. C. Spencer, On deformations of complex analytic structure, III, stability theorems for complex structures, Ann. of Math., 71 (1960), 43–76.
  • Z. J. Liu and P. Xu, On quadratic Poisson structures, Lett. Math. Phys., 26 (1992), 33–42
  • Z. J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547–574.
  • Y. Namikawa, Poisson deformations of affine symplectic varieties, Math. AG/0609741
  • A. Polishchuk, Algebraic geometry of Poisson varieties, J. Math. Sci. (N.Y.), 84 (1997), 1413–1444.
  • F. Sakai, Anti-Kodaira dimension of ruled surfaces, Sci. Rep. Saitama Univ., 2 (1982), 1–7.
  • J. P. Serre, Lie Algebra and Lie groups, Lecture Notes in Math. 1500, Springer-Verlag.
  • G. Tian, Smoothness of the universal deformations spaces of compact Calabi-Yau manifolds and its Peterson-Weil metric, (ed. S. T. Yau) Mathematical Aspect of string theory, World Scientific Publishing co., Singapole, 1987, pp.,629–646.
  • I. Vaisman, Reduction and submanifolds of generalized complex manifolds, Differential Geom. Appl., 25 (2007), 147–166.