# Picard group and quantization of toric orbifolds

## TL;DR

This paper provides a combinatorial description of the orbifold Picard group for complete toric orbifolds, elucidating the complex-symplectic relationship and illustrating the breakdown of line bundle identification via Chern classes.

## Contribution

It introduces a combinatorial framework for the orbifold Picard group, revealing differences from the smooth case and exploring implications for symplectic reduction and Bohr--Sommerfeld conditions.

## Key findings

- Breakdown of line bundle identification by Chern class in orbifolds
- Non-constancy of $h^0$ on same Chern class representatives
- Application to symplectic reduction and mixed polarizations

## Abstract

In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a combinatorial description of the orbifold Picard group for complete toric orbifolds, with the aim of detailing the relation between complex and symplectic aspects in the orbifold setting. In particular this permits to illustrate the breakdown of identification of (orbifold) line bundles by their Chern class (or moment polytope up to translations in $\mathfrak{t}^\ast$), and non-constancy of $h^0$ on representatives of the same Chern class. As an application, we discuss symplectic reduction with respect to restrictions of the action to sub-tori, and the associated Bohr--Sommerfeld conditions in mixed polarizations.