{
  "id": "1407.4223",
  "version": "v2",
  "published": "2014-07-16T07:55:57.000Z",
  "updated": "2014-07-17T10:44:31.000Z",
  "title": "GIT characterizations of Harder-Narasimhan filtrations",
  "authors": [
    "Alfonso Zamora"
  ],
  "comment": "Ph.D. Thesis, Universidad Complutense de Madrid, defended on June 2013. Published on the e-prints server of the University on July 2013, http://eprints.ucm.es/22525/ Original manuscript with references of recent related works added",
  "categories": [
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-17T10:44:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harder-narasimhan filtration",
      "git characterizations",
      "torsion free coherent sheaves",
      "maximal unstability",
      "comprising finite dimensional quiver representations"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4223Z"
    }
  }
}