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Twistor Hecke eigensheaves in genus 2

Ron Donagi, Tony Pantev, Carlos Simpson

Abstract

Following the strategy outlined in [DP09] arXiv:math/0604617 and [DP22] arXiv:math/0604617 for bundles of rank 2 on a smooth projective curve of genus $2$, we construct flat connections over the moduli of stable bundles, with singularities along the wobbly locus. We verify that the associated $D$-modules are Hecke eigensheaves. The local systems are constructed by the nonabelian Hodge correspondence from Higgs bundles. The spectral varieties of the Higgs bundles are the Hitchin fibers corresponding to the Hecke eigenvalues.

Twistor Hecke eigensheaves in genus 2

Abstract

Following the strategy outlined in [DP09] arXiv:math/0604617 and [DP22] arXiv:math/0604617 for bundles of rank 2 on a smooth projective curve of genus , we construct flat connections over the moduli of stable bundles, with singularities along the wobbly locus. We verify that the associated -modules are Hecke eigensheaves. The local systems are constructed by the nonabelian Hodge correspondence from Higgs bundles. The spectral varieties of the Higgs bundles are the Hitchin fibers corresponding to the Hecke eigenvalues.
Paper Structure (111 sections, 166 theorems, 973 equations, 2 figures)

This paper contains 111 sections, 166 theorems, 973 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric Langlands
  3. The case of genus 2 and rank 2
  4. Main theorems
  5. Structure of the paper
  6. Acknowledgements
  7. Synthetic approach
  8. Pencils of quadrics in P5
  9. Basic geometry of the lines
  10. The wobbly divisor
  11. Synthetic correspondences and rigidification
  12. The quadric line complex
  13. Hecke curves and Kummer surfaces
  14. General considerations
  15. Curves
  16. ...and 96 more sections

Key Result

Theorem 1.1

The base affine spaces of the Hitchin fibration ${\bf B}$ for $G$ and ${}^LG$ are naturally isomorphic. Furthermore, the two Hitchin maps $\boldsymbol{h}$ and ${}^L\boldsymbol{h}$ are generically dual SYZ-type torus fibrations.

Figures (2)

  • Figure 1: The Hirzebruch surface ${\overline{\mathcal{H}}}(a)_{\ell}$
  • Figure 2: Divisor configuration after two blowups

Theorems & Definitions (323)

  • Theorem 1.1
  • Conjecture 1.2: DP1DonagiPantev
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 313 more