Non-simplicial quantum toric varieties
Antoine Boivin
TL;DR
The paper generalizes quantum toric varieties beyond simplicial, rational fans by introducing presented calibrated quantum tori and a per-cone calibration $\varphi$, proving an equivalence between calibrated quantum fans and calibrated quantum toric varieties. It builds affine local models for maximal cones, defines global glued objects, and establishes a GIT-like quotient description while highlighting obstructions to a Gale-transform realization in the non-simplicial setting. The framework yields a gerbe structure when calibration data is forgotten, revealing a stacky, torus-fibered relationship between calibrated and non-calibrated variants. Overall, it extends VQS to arbitrary fans, clarifying local models, morphisms, and global realizations in the non-simplicial case while preserving stack-theoretic and gerbe features.
Abstract
In this paper, we define quantum toric varieties associated to an arbitrary fan in a finitely generated subgroup of some $\mathbb{R}^d$ generalizing the article arXiv:2002.03876 of Katzarkov, Lupercio, Meersseman and Verjovsky.