The lattice points of a stacky polytope

TL;DR

This paper explores the connection between lattice points of stacky polytopes and prequantisation of associated Deligne--Mumford stacks, providing explicit conditions and discussing the related line bundles.

Contribution

It introduces a notion of prequantisation for stacks, establishes conditions for existence, and links lattice points to holomorphic line bundles in this context.

Findings

Explicit conditions for prequantisation existence in terms of stacky polytopes

Introduction of a prequantisation concept for Deligne--Mumford stacks

Relation between lattice points and holomorphic line bundles

Abstract

We study the relation between the lattice points of a stacky polytope and a prequantisation of the stack associated to the stacky polytope. We introduce a prequantisation of a Deligne--Mumford stack and discuss the uniqueness and the existence of a prequantisation. After that we describe explicitly the condition for the existence of a prequantisation in terms of stacky polytope under some conditions and discuss the relation between the holomorphic line bundle associated to a prequantisation and the lattice points of the stacky polytope.