A guide to moduli theory beyond GIT

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.