Moduli spaces of quantum toric stacks and their compactification

TL;DR

This paper explores the moduli spaces and compactification of quantum toric stacks, which generalize classical toric varieties by replacing lattices with finitely generated subgroups, addressing their rigidity issues.

Contribution

It introduces the moduli spaces of quantum toric stacks and develops their compactification, extending the theory of toric varieties to a stacky, quantum setting.

Findings

Construction of moduli spaces for quantum toric stacks

Development of a compactification method for these moduli spaces

Extension of classical toric variety theory to quantum stacks

Abstract

A toric variety is a normal complex variety which is completely described by combinatorial data, namely by a fan of strongly convex rational (with respect to a lattice) cones. Due to this rationality condition, toric varieties are (equivariantly) rigid since a deformation of the lattice can make it dense. A solution to this problem consists in considering quantum toric stacks. The latter is a stacky generalization of toric varieties where the "lattice" is replaced by a finitely generated subgroup of $\mathbb{R}^d$ (in the simplicial case as introduced by L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky). The goal of this paper is to explain the moduli spaces of quantum toric stacks and their compactification.