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

Picard group of isotropic realizations of twisted Poisson manifolds

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.
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

[14]

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

[15]

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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