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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
References:
[1]

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

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.  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), 267-310. doi: 10.1007/s00205-012-0512-9.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, 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]

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

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. 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), 377-412. doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

show all references

References:
[1]

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

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.  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), 267-310. 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), 251-256. 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), 549-607. 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), 39-66, 310.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles, 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]

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

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. 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), 377-412. doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

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