Abstract
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 `Neron-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.
Highlights
80s, where he worked in the slightly more general setting of Lagrangian fiber bundles
Dazord and Delzant in [DD] studied, in the spirit of [Dui], the global structure of symplectically complete isotropic realizations (SCIR) which correspond to superintegrable systems
Sansonetto and Sepe, in the recent paper [SS], generalized the Dazord-Delzant theory in both contexts of superintegrable systems and almost symplectic systems. They studied what we call in this paper almost symplectically complete isotropic realizations (ASCIR, see Definition 2.8), which are equivalent to IASHSs
Summary
In [FS], the notion of strong Hamiltonianicity is introduced and exploited to generalize the Liouville-Arnold theorem on local action-angle coordinates in the context of integrable almost symplectic Hamiltonian systems (cf. Definition 2.1), which are mechanical systems that, in terms of generality, lie between Hamiltonian ones and nonholonomic ones (cf. [SS]). (4) The almost symplectic leaves are locally the level sets of the action coordinates a1, · · · , an on B, which are locally the Casimirs of the twisted Poisson structure of B. (5) The local action coordinates give rise to a tropical affine structure transversal to the almost symplectic leaves of B. In Theorem 2.7, we start with the almost symplectic and strongly Hamiltonian structures of the integrable system M , and deduce the various compatible structures of B, i.e. the regular twisted Poisson structure and the tropical affine structure. This is the geometric mechanical approach taken in [FS]. Proposition 3.8. ([SS, Proposition 4]) P and the tropical affine structure give rise to each other
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
