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

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 Kollár’s condition and the category of multi-uniformized twisted varieties, and we construct the corresponding moduli space. We then broaden the framework to encompass Kollár’s Seifert ${\mathbb{G}}_m^d$-bundles, showing that their moduli likewise coincide with those of $d$-uniformized $d$-cyclotomic orbispaces.