Picard group of isotropic realizations of twisted poisson manifolds

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 [DD]. 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 ‘N´eron-Severi group’ is isomorphic to H(2) (B,P). An example of an ASCIR over a certain open subset of a compact Lie group is discussed.