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June  2016, 8(2): 179-197. doi: 10.3934/jgm.2016003

Picard group of isotropic realizations of twisted Poisson manifolds

Chi-Kwong Fok 1,

1. 

Department of Mathematics, Cornell University, Ithaca, NY 14853, United States

Received  March 2015 Revised  December 2015 Published  June 2016

Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of [10]. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `Néron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.
Keywords: Picard group., Dazord-Delzant homomorphism, twisted Poisson manifold, Almost symplectically complete isotropic realization, tropical affine structure.
Mathematics Subject Classification: Primary: 53D15, 53D17; Secondary: 55N3.
Citation: Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003
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show all references

References:
[1]

in Asterique, 327 (2009), 131-199.  Google Scholar

[2]

J. Differential Geom., 48 (1998), 445-495.  Google Scholar

[3]

Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

C. R. Acad. Sci. Paris, 336 (2003), 251-256. doi: 10.1016/S1631-073X(02)00024-9.  Google Scholar

[5]

Duke Math. J., 123 (2004), 549-607. doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[6]

Moscow Math. J., 4 (2004), 39-66, 310.  Google Scholar

[7]

J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[8]

Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[9]

J. Differential Geom., 66 (2004), 71-137.  Google Scholar

[10]

J. Differential Geom., 26 (1987), 223-251.  Google Scholar

[11]

H. Duan, Schubert calculus and cohomology of Lie groups. Part II. Compact Lie groups,, preprint, ().   Google Scholar

[12]

Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.  Google Scholar

[13]

J. Math. Phys., 48 (2007), 092902, 13pp. doi: 10.1063/1.2783937.  Google Scholar

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Duke Math. J., 89 (1997), 377-412. doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

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Graduate Text in Mathematics, 52, Springer-Verlag, 1977.  Google Scholar

[16]

J. Geometric Mechanics, 5 (2013), 233-256. doi: 10.3934/jgm.2013.5.233.  Google Scholar

[17]

in Noncommutative Geometry and String Theory (Yokohama, 2001), ed. Y. Maeda and S. Watamura, Progr. Theoret. Phys. Suppl. 144, Kyoto Univ., Kyoto, 2001, 145-154. doi: 10.1143/PTPS.144.145.  Google Scholar

[18]

Bull. Braz. Math. Soc., 42 (2011), 783-803. doi: 10.1007/s00574-011-0035-2.  Google Scholar

[19]

Progress in Mathematics, Birhaüser Verlag, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[20]

in Symplectic geometry, groupoids, and integrable systems, Séminaire sud-rhodanien de géométrie à Berkeley, Math. Sci. Res. Inst. Publ., 20, Springer-Verlag, New York, 1991, 291-311. doi: 10.1007/978-1-4613-9719-9_20.  Google Scholar

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