## Journal of the Mathematical Society of Japan

### Poisson structures and generalized Kähler submanifolds

Ryushi GOTO

#### 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

http://projecteuclid.org/euclid.jmsj/1234189030

Digital Object Identifier
doi:10.2969/jmsj/06110107

Mathematical Reviews number (MathSciNet)
MR2272873

#### Citation

GOTO, Ryushi. Poisson structures and generalized Kähler submanifolds. J. Math. Soc. Japan 61 (2009), no. 1, 107--132. doi:10.2969/jmsj/06110107. http://projecteuclid.org/euclid.jmsj/1234189030.

#### 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.