Peng Zhou

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====== Hitchin Fibration – Learning Roadmap ======

The Hitchin fibration is one of the central structures in modern geometry.  
It connects:

  * Higgs bundles
  * integrable systems
  * non-abelian Hodge theory
  * geometric Langlands
  * mirror symmetry

This page collects a recommended path to learn the subject, with references.

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===== 1. Hitchin's Original Paper =====

The starting point of the whole subject.

**Nigel Hitchin (1987)**  
"Stable bundles and integrable systems"

Introduces:

  * Higgs bundles
  * the Hitchin map
  * the integrable system structure
  * the Morse function \(f = |\phi|^2\)

Link:  
https://doi.org/10.1215/S0012-7094-87-05408-1

PDF:  
https://people.maths.ox.ac.uk/hitchin/hitchin87.pdf

Key ideas to focus on:

  * definition of Higgs bundle
  * Hitchin base
  * spectral curves
  * Morse theory on the moduli space

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===== 2. Spectral Curve Description (BNR) =====

This explains the **spectral correspondence**.

**Beauville – Narasimhan – Ramanan**

"Spectral curves and the generalized theta divisor"

Link:  
https://math.univ-cotedazur.fr/~beauvill/pubs/bnr.pdf

The main result:

(E, φ) ↔ line bundle on spectral curve

Consequences:

  * Hitchin fibers become Jacobians
  * Hitchin system becomes an algebraically completely integrable system.

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===== 3. Non-abelian Hodge Theory =====

**Carlos Simpson (1992)**  
"Higgs bundles and local systems"

Link:  
https://www.numdam.org/item/PMIHES_1992__75__5_0.pdf

Explains the correspondence:

Higgs bundles  
↕  
flat connections  
↕  
representations of π₁

Important features:

  * harmonic metrics
  * C* action on Higgs moduli
  * fixed points (VHS)
  * Morse theory of Hitchin function

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===== 4. Geometry of the Nilpotent Cone =====

**Gérard Laumon**

{{ :notes:laumon_nilpotent_cone.pdf |"Un analogue global du cône nilpotent"}}


Main result:

  * the global nilpotent cone is a **Lagrangian substack** of the cotangent stack of Bun.

Key ideas:

  * stratification by nilpotent type
  * relation to Harder–Narasimhan strata
  * geometry of Hitchin fiber over 0

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===== 5. Modern Survey =====

**Hausel – Thaddeus**

"Mirror symmetry, Langlands duality, and the Hitchin system"

Link:  
https://arxiv.org/abs/math/0205236

Very good conceptual overview of:

  * Hitchin integrable systems
  * mirror symmetry
  * Langlands duality
  * topology of moduli spaces

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===== 6. Ngô and the Global Geometry =====

**Ngô Bao Châu**

Work on the Hitchin fibration used to prove the Fundamental Lemma.

Example reference:

https://arxiv.org/pdf/0801.0446

Key themes:

  * geometry of singular Hitchin fibers
  * support theorem
  * role of Hitchin fibration in geometric Langlands

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===== 7. Lecture Notes / Friendly Introductions =====

**Tamás Hausel – Hitchin systems**

https://hausel.pages.ist.ac.at/wp-content/uploads/sites/229/2024/09/gths.pdf

Good for:

  * topology of Higgs moduli
  * Morse theory
  * integrable system viewpoint

-----

Pavel Etingof and Henry Liu, lecture at BIMSA


https://arxiv.org/pdf/2409.09505

 

===== 8. Topics to Understand =====

Important themes when studying the Hitchin fibration:

  * spectral curves
  * integrable system structure
  * C* action on Higgs moduli
  * Morse theory of f = |φ|²
  * nilpotent cone
  * singular Hitchin fibers

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