# git & nahc

This is where Enya Hsiao and I organise the *Geometric Invariant Theory & Non-abelian Hodge Correspondence* Reading Seminar.

Here is the current version of the seminar plan!

**Update 2024-04-09:** Final update, all notes uploaded! Thank you all for participating!

The nonabelian Hodge correspondence is a deep and beautiful theory that identifies three moduli spaces: the **Betti moduli space** \(\mathcal{M}_{\text{B}}\) of surface group representations, the **de Rham moduli space** \(\mathcal{M}_{\text{dR}}\) of flat connections and the **Dolbeault moduli space** \(\mathcal{M}_{\text{Dol}}\) of Higgs bundles, whose objects on the surface appear to arise from very different contexts. The moduli spaces themselves are constructed as GIT quotients based on the geometric invariant theory pioneered by Mumford. The correspondence between these moduli spaces is the cumulative work of Corlette, Donaldson, Hitchin and Simpson built on that of many mathematicians and the machinery involved to prove these results lies in the intersection of algebraic geometry, complex geometry, geometric analysis and gauge theory.

The goal of this reading seminar is two-fold:

- Understand the workings of geometric invariant theory and how they are applied in the construction of the moduli spaces involved in the nonabelian Hodge correspondence.
- Understand the geometric properties of objects in the moduli spaces and without going into too much nasty detail, sketch the correspondence between the moduli spaces for \(G = \mathrm{GL}(n, \mathbb{C})\).

## Schedule

Date | Topic | Speaker | |
---|---|---|---|

16.11. | 0. | Introduction and overview | |

23.11. | 1. | Affine and projective varieties | Leonie |

30.11. | 2. | Algebraic groups and invariant theory | Barbara |

07.12. | 3. | Affine GIT | Anaëlle |

14.12. | D1. | Discussion: Gauge Theory | Merik |

21.12. | ❄️ | ||

28.12. | ❄️ | ||

04.01. | ❄️ | ||

11.01. | 4. | The Betti Moduli space: Character variety | Fernando |

18.01. | 5. | The de Rham Moduli space: Flat connections | Jiajun |

25.01. | 6. | The Riemann-Hilbert correspondence | Pengfei |

02.02. | D2. | Discussion: Proj and Ample line bundles 14:00 | Bernhard |

08.02. | 7. | Projective GIT | Bernhard |

15.02. | 8. | Hilbert-Mumford criterion | Maximilian |

22.02. | 9. | The moduli problem | Leonie |

04.03. | D3. | Discussion: Complex geometry 11:00 | Enya |

07.03. | 🏕️ | ||

14.03. | 10. | The Dolbeault Moduli space: Higgs bundles Room E2 10 | Tim |

21.03. | 11. | Harmonic bundles and Donaldson-Corlette correspondence | Christian |

28.03. | 12. | The Hitchin-Simpson correspondence | Enya |

04.04. | D4. | Discussion: The rank one case | Enya & Tim |