GIT characterizations of Harder-Narasimhan filtrations

TL;DR

This thesis explores the relationship between GIT maximal unstability and Harder-Narasimhan filtrations across various moduli problems, establishing a correspondence between GIT 1-parameter subgroups and filtrations.

Contribution

It demonstrates a general correspondence between GIT maximal unstability and Harder-Narasimhan filtrations for multiple moduli problems, extending previous specific cases.

Findings

Established a correspondence for torsion free sheaves, quiver representations, and tensors.

Connected Hesselink stratification with Harder-Narasimhan types.

Unified GIT and Harder-Narasimhan concepts across different moduli problems.

Abstract

This Ph.D. thesis studies the relation between the Harder-Narasimhan filtration and a notion of GIT maximal unstability. When constructing a moduli space by using Geometric Invariant Theory (GIT), a notion of GIT stability appears, which is determined by 1-parameter subgroups. This thesis shows a correspondence between the 1-parameter subgroup giving maximal unstability from the GIT point of view and the Harder-Narasimhan filtration for different moduli problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank 2 tensors and quiver representations. The article [GSZ] contains the correspondence for torsion free coherent sheaves, whereas [Za1] and [Za2] are devoted to finite dimensional quiver representations and rank 2 tensors. In [HK], the authors explore this kind of correspondences while identifying the Hesselink stratification on conjugacy classes of 1-parameter subgroups with the one on Harder-Narasimhan types, showing that the Hesselink's adapted 1-parameter subgroup corresponds to the Harder-Narasimhan filtration (which is previously prescribed on each strata). Futher work of Hoskins (c.f. [Ho1, Ho2]) continues on the same direction for other moduli problems, comprising finite dimensional quiver representations as it is done in [Za1], and coherent sheaves using the functorial construction of Alvarez-Consul and King (c.f. [ACK]), as it appears in Section 3.2 of this thesis.