Generalizing the GAGA Principle

TL;DR

This paper extends the GAGA principle to non-separated spaces, stacks, and families, demonstrating that certain analytic compactifications of moduli spaces are algebraizable, thus broadening the principle's applicability.

Contribution

It generalizes the classical GAGA results to more complex geometric contexts, including stacks and families, and applies these to algebraize specific moduli space compactifications.

Findings

GAGA results extended to non-separated spaces

GAGA results extended to stacks

Analytic compactifications of moduli spaces are algebraizable

Abstract

This paper generalizes the fundamental GAGA results of Serre cite{MR0082175} in three ways---to the non-separated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of $\mathcal{M}_{g,n}$ possessing modular interpretations are algebraizable.