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Spectral Covers

Ron Donagi

Abstract

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

Spectral Covers

Abstract

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

Paper Structure

This paper contains 15 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Hitchin's system
  3. Some related systems
  4. Decomposition of spectral Picards
  5. The question
  6. Decomposition of spectral covers
  7. Decomposition of spectral Picards
  8. The distinguished Prym
  9. Abelianization
  10. Abstract vs. $K$-valued objects
  11. The regular semisimple case: the shift
  12. The regular case: the twist along the ramification
  13. Adding values and representations
  14. Symplectic and Poisson structures
  15. Some applications and problems

Key Result

Theorem 1

The cotangent bundle $T^*{\cal M}$ is an algebraically completely integrable Hamiltonian system.

Theorems & Definitions (14)

  • Theorem 1: HitchinH1
  • Proposition 2
  • Example 3
  • Example 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Example 9
  • Proposition 10
  • ...and 4 more