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), 131-199.

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251-256. doi: 10.1016/S1631-073X(02)00024-9.

[5]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. doi: 10.1215/S0012-7094-04-12335-8.

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39-66, 310.

[7]

A. Catteneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.

[8]

M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.

[11]

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

[12]

J. J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. doi: 10.1063/1.2783937.

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377-412. doi: 10.1215/S0012-7094-97-08917-1.

[15]

R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, 52, Springer-Verlag, 1977.

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures, J. Geometric Mechanics, 5 (2013), 233-256. doi: 10.3934/jgm.2013.5.233.

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background, 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.

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry, Bull. Braz. Math. Soc., 42 (2011), 783-803. doi: 10.1007/s00574-011-0035-2.

[19]

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

[20]

P. Xu, Morita equivalent symplectic groupoids, 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.

show all references

References:
[1]

A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups, in Asterique, 327 (2009), 131-199.

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251-256. doi: 10.1016/S1631-073X(02)00024-9.

[5]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607. doi: 10.1215/S0012-7094-04-12335-8.

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39-66, 310.

[7]

A. Catteneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.

[8]

M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.

[11]

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

[12]

J. J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. doi: 10.1063/1.2783937.

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377-412. doi: 10.1215/S0012-7094-97-08917-1.

[15]

R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, 52, Springer-Verlag, 1977.

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures, J. Geometric Mechanics, 5 (2013), 233-256. doi: 10.3934/jgm.2013.5.233.

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background, 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.

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry, Bull. Braz. Math. Soc., 42 (2011), 783-803. doi: 10.1007/s00574-011-0035-2.

[19]

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

[20]

P. Xu, Morita equivalent symplectic groupoids, 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.

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