A guide to moduli theory beyond GIT

TL;DR

This survey explores recent stack-theoretic methods for constructing moduli spaces, introduces $ heta$-stratifications, and applies these ideas to principal bundles and their compactifications on higher-dimensional varieties.

Contribution

It provides an overview of stack-based moduli space construction techniques, introduces $ heta$-stratifications, and applies them to principal bundles and $ ho$-sheaves, advancing the understanding of moduli problems.

Findings

Development of $ heta$-stratifications for stacks

Construction of stratifications by instability types

Proper moduli spaces for Gieseker semistable objects

Abstract

In this survey we provide an overview of some recent developments in the construction of moduli spaces using stack-theoretic techniques. We will also explain the analogue of Harder-Narasimhan stratifications for general stacks, known as $\Theta$-stratifications. As an application of the ideas exposed here, we address the moduli problem of principal bundles over higher dimensional projective varieties, as well as its different compactifications by the so-called principal $\rho$-sheaves. We construct a stratification by instability types whose lower strata admits a proper good moduli space of ``Gieseker semistable" objects and a new Gieseker-type Harder-Narasimhan filtration for these objects. Detailed proofs of the latter results will appear elsewhere.