# American Institute of Mathematical Sciences

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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##### References:
 [1] Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233 [2] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [3] Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61 [4] Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008 [5] Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385 [6] Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155 [7] Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173 [8] Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37 [9] Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453 [10] Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56. [11] Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2685-2702. doi: 10.3934/jimo.2020089 [12] Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032 [13] Nuh Aydin, Nicholas Connolly, Markus Grassl. Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Advances in Mathematics of Communications, 2017, 11 (1) : 245-258. doi: 10.3934/amc.2017016 [14] Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939 [15] Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267 [16] Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291 [17] Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109 [18] Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282 [19] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 [20] Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220

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