Picard group of isotropic realizations of twisted Poisson manifolds

TL;DR

This paper classifies isotropic realizations of twisted Poisson manifolds, introduces a Picard group structure, and describes it using sheaf cohomology, with applications to Lie groups.

Contribution

It constructs the Picard group for twisted Poisson manifolds and describes its structure via sheaf cohomology, extending the understanding of isotropic realizations.

Findings

The Picard group is described by exact sequences involving sheaf cohomology.

The Néron-Severi group is isomorphic to H^2(B, P).

An example over a compact Lie group illustrates the theory.

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 \cite{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 $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.