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
References:
[1]

A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups,, in Asterique, 327 (2009), 131.   Google Scholar

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps,, J. Differential Geom., 48 (1998), 445.   Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups,, C. R. Acad. Sci. Paris, 336 (2003), 251.  doi: 10.1016/S1631-073X(02)00024-9.  Google Scholar

[5]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry,, Moscow Math. J., 4 (2004), 39.   Google Scholar

[7]

A. Catteneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[8]

M. Crainic and R. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles,, J. Differential Geom., 26 (1987), 223.   Google Scholar

[11]

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

[12]

J. J. Duistermaat, On global action-angle variables,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles,, Duke Math. J., 89 (1997), 377.  doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

R. Hartshorne, Algebraic Geometry,, Graduate Text in Mathematics, 52 (1977).   Google Scholar

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures,, J. Geometric Mechanics, 5 (2013), 233.  doi: 10.3934/jgm.2013.5.233.  Google Scholar

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background,, in Noncommutative Geometry and String Theory (Yokohama, (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry,, Bull. Braz. Math. Soc., 42 (2011), 783.  doi: 10.1007/s00574-011-0035-2.  Google Scholar

[19]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[20]

P. Xu, Morita equivalent symplectic groupoids,, in Symplectic geometry, 20 (1991), 291.  doi: 10.1007/978-1-4613-9719-9_20.  Google Scholar

show all references

References:
[1]

A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups,, in Asterique, 327 (2009), 131.   Google Scholar

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps,, J. Differential Geom., 48 (1998), 445.   Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups,, C. R. Acad. Sci. Paris, 336 (2003), 251.  doi: 10.1016/S1631-073X(02)00024-9.  Google Scholar

[5]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Math. J., 123 (2004), 549.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry,, Moscow Math. J., 4 (2004), 39.   Google Scholar

[7]

A. Catteneo and P. Xu, Integration of twisted Poisson structures,, J. Geom. Phys., 49 (2004), 187.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[8]

M. Crainic and R. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles,, J. Differential Geom., 26 (1987), 223.   Google Scholar

[11]

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

[12]

J. J. Duistermaat, On global action-angle variables,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles,, Duke Math. J., 89 (1997), 377.  doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

R. Hartshorne, Algebraic Geometry,, Graduate Text in Mathematics, 52 (1977).   Google Scholar

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures,, J. Geometric Mechanics, 5 (2013), 233.  doi: 10.3934/jgm.2013.5.233.  Google Scholar

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background,, in Noncommutative Geometry and String Theory (Yokohama, (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry,, Bull. Braz. Math. Soc., 42 (2011), 783.  doi: 10.1007/s00574-011-0035-2.  Google Scholar

[19]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[20]

P. Xu, Morita equivalent symplectic groupoids,, in Symplectic geometry, 20 (1991), 291.  doi: 10.1007/978-1-4613-9719-9_20.  Google Scholar

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