Quantum Multiplicative Hypertoric Varieties and Localization

TL;DR

This paper develops a theory of q-deformed multiplicative hypertoric varieties, constructs associated algebras of q-difference operators, and establishes their Azumaya properties and localization at roots of unity.

Contribution

It introduces a new framework for q-deformations of hypertoric varieties, constructs a Heisenberg double algebra, and proves Azumaya splitting and localization results at roots of unity.

Findings

Construction of algebra Dq as a Heisenberg double

Azumaya algebra on l-twisted hypertoric variety at roots of unity

Splitting of algebra over fibers of moment and resolution maps

Abstract

We consider q-deformations of multiplicative hypertoric varieties, for q a non-zero element of an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumaya algebras.