A guide to moduli theory beyond GIT (Q6607622)

In this survey paper, the authors present a summary of recent advances in the construction of moduli spaces through stack-theoretic methods using \(\Theta\)-stratifications rather than Harder-Narasimhan stratifications. As an application of these concepts, they tackle the moduli problem for principal bundles on higher-dimensional projective varieties and explore their compactifications via so-called principal \(\rho\)-sheaves.\N\NIn section 2, the authors give an intrinsic approach to moduli problems. They also develop a stratification based on instability types, where the lower strata provide a proper good moduli space for ``Gieseker semistable'' objects, along with a new Gieseker-type Harder-Narasimhan filtration for these objects. In section 3, they consider one of the key examples in moduli theory: the moduli problem for torsion-free sheaves, applying the previous concepts to this example. In the last section, the authors deal with the moduli of \(G\)-bundles on higher-dimensional varieties.\N\NFor the entire collection see [Zbl 1545.14003].