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Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves

Philip Boalch

TL;DR

This work surveys how the nonabelian Hodge picture on curves extends to irregular (meromorphic) curves by introducing irregular types and parabolic data, leading to moduli spaces of meromorphic connections and Higgs bundles that carry complete hyperkähler metrics. The irregular framework unifies three viewpoints—Dolbeault, De Rham, and Betti—via generalized Riemann–Hilbert correspondences and Hitchin-type integrable systems, and it links to Fourier–Laplace transforms, Baker–Akhiezer functions, and Frobenius-manifold invariants. Concrete results include the hyperkähler structure on irregular moduli, additive and ADE-type 2-dimensional examples related to Painlevé equations, and a detailed E8 construction connecting Dolbeault, De Rham, additive, and Betti spaces. The study highlights how irregular data diversify the landscape of wild character varieties and open pathways for new braid-type symmetries and isomonodromic dynamics within a broad geometric framework.

Abstract

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves

TL;DR

This work surveys how the nonabelian Hodge picture on curves extends to irregular (meromorphic) curves by introducing irregular types and parabolic data, leading to moduli spaces of meromorphic connections and Higgs bundles that carry complete hyperkähler metrics. The irregular framework unifies three viewpoints—Dolbeault, De Rham, and Betti—via generalized Riemann–Hilbert correspondences and Hitchin-type integrable systems, and it links to Fourier–Laplace transforms, Baker–Akhiezer functions, and Frobenius-manifold invariants. Concrete results include the hyperkähler structure on irregular moduli, additive and ADE-type 2-dimensional examples related to Painlevé equations, and a detailed E8 construction connecting Dolbeault, De Rham, additive, and Betti spaces. The study highlights how irregular data diversify the landscape of wild character varieties and open pathways for new braid-type symmetries and isomonodromic dynamics within a broad geometric framework.

Abstract

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

Paper Structure

This paper contains 14 sections, 3 theorems, 30 equations, 2 figures.

Table of Contents

  1. Big picture
  2. Irregular curves
  3. Flags/filtrations
  4. Irregular connections
  5. Irregular Higgs bundles
  6. More motivation
  7. Fourier--Laplace
  8. Baker functions
  9. Nonabelian Hodge theoretic invariants
  10. Examples of moduli spaces
  11. Additive approximations
  12. Low dimensional examples
  13. $E_8$ examples
  14. Conclusion

Key Result

Lemma 1

Hint: use the valuative criterion for properness for the compositions $\mathbb{C}^* \hookrightarrow \mathcal{M}_{\text{\rm B}}\to \mathcal{M}_{\text{\rm DR}}\to \text{\rm Jac}(\Sigma)$. There is no algebraic isomorphism $\mathcal{M}_{\text{\rm B}} \to \mathcal{M}_{\text{\rm DR}}$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (6)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5
  • Proposition 6