Proper splittings and projectivity for good moduli spaces

TL;DR

This paper proves that good moduli spaces can be split after certain coverings and extends Kollár's ampleness lemma to provide a criterion for their projectivity.

Contribution

It introduces a method to split good moduli spaces via proper coverings and generalizes Kollár's ampleness lemma for establishing projectivity criteria.

Findings

Existence of splittings after proper, generically finite coverings.

Generalization of Kollár's ampleness lemma.

New criterion for projectivity of good moduli spaces.

Abstract

We show that any good moduli space $\pi : \mathcal{X} \to Y$ has a splitting after a proper, generically finite covering of $Y$. As an application we generalize Koll\'ar's ampleness lemma to give a criterion for projectivity of a good moduli space.