The moduli stack of principal \(\rho \)-sheaves and Gieseker-Harder-Narasimhan filtrations (Q6562903)

An interesting topic in algebraic geometry is the construction of moduli spaces parametrizing certain objects: for instance, there exist moduli spaces of curves, bundles, \(G\)-principal bundles, and decorated bundles. Most of these constructions are carried out using techniques from Geometric Invariant Theory (GIT). However, there exists another approach to constructing these moduli spaces by means of algebraic stacks (see [\textit{J. Alper} et al., Invent. Math. 234, No. 3, 949--1038 (2023); Zbl 1541.14018]). In particular, the authors of this paper follow the steps devised in [\textit{D. Halpern-Leistner}, ``On the structure of instability in moduli theory,'' Preprint, \url{arXiv:1411.0627}; \textit{D. Halpern-Leistner} et al., J. Reine Angew. Math. 809, 159--215 (2024; Zbl 07829701)] to construct the moduli space of semistable \(\rho\)-sheaves over a smooth projective variety, where \(\rho\) is a faithful representation of a connected reductive group \(G\) into a product of general linear groups. This moduli space was previously constructed by the first author and some collaborators using methods from GIT.\N\NOne of the main results of this paper is that the moduli stack \(\mathrm{Bun}_\rho(X)\) of \(\rho\)-sheaves admits a \(\Theta\)-stratification induced by a polynomial numerical invariant, and fixing a tuple of rational polynomials \(P^\bullet\), the open substack \(\mathrm{Bun}_\rho(X)^{ss,P^\bullet}\) of \(\rho\)-sheaves with tuple \(P^\bullet\) as Hilbert polynomials admits a proper good moduli space. On the other hand, another result is the analogous of the filtration on unstable bundles (known as the Harder-Narasimhan filtration). They obtain a uniquely multi-weighted filtration on every principal \(\rho\)-sheaf.\N\NThe article is well-structured and very well presented. It effectively introduces the motivation behind the study and provides a clear context for what will be developed. The author also relates the results to existing literature, drawing connections with other ones, some of which have been obtained in different contexts. Furthermore, the article includes an appendix where the GHN-filtration is discussed in positive characteristic.