Moduli of Multi-Uniformized Stacks and Seifert $\mathbb{G}_m^d$-Bundles

TL;DR

This paper develops a new framework for multi-uniformized stacks and Seifert $ abla$-bundles, establishing their moduli spaces and equivalences with certain categories of line bundles and orbispaces.

Contribution

It introduces multi-uniformized stacks, proves their equivalence with multi $bQ$-line bundles, and constructs their moduli spaces, extending to Seifert $bG_m^d$-bundles.

Findings

Established an equivalence between multi $bQ$-line bundles and multi-uniformized twisted varieties.

Constructed the moduli space for multi-uniformized stacks.

Extended the framework to include Kollár's Seifert $bG_m^d$-bundles and their moduli.

Abstract

We introduce multi-uniformized stacks as a generalization of the Abramovich--Hassett construction of uniformized twisted varieties. We prove an equivalence between the category of multi $\mathbb{Q}$-line bundles satisfying an analogue of Koll\'{a}r's condition and the category of multi-uniformized twisted varieties, and we construct the corresponding moduli space. We then broaden the framework to encompass Koll\'{a}r's Seifert $\mathbb{G}_m^d$-bundles, showing that their moduli likewise coincide with those of $d$-uniformized $d$-cyclotomic orbispaces.