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Higgs bundles twisted by a vector bundle

Guillermo Gallego, Oscar Garcia-Prada, M. S. Narasimhan

Abstract

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin-Kobayashi correspondence for a generalization of the Hitchin equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher dimensional variety.

Higgs bundles twisted by a vector bundle

Abstract

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin-Kobayashi correspondence for a generalization of the Hitchin equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher dimensional variety.

Paper Structure

This paper contains 20 sections, 16 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. -twisted Higgs bundles
  3. The main definitions
  4. The moduli space
  5. The moduli functor
  6. $V$-twisted Higgs bundles as sheaves on $V$
  7. Construction of the moduli space
  8. $S$-equivalence and polystability
  9. Infinitesimal deformations
  10. The Hitchin morphism for -twisted -Higgs bundles
  11. The Hitchin map
  12. Universal spectral data
  13. The spectral correspondence
  14. Properties of the spectral curve
  15. Integrality
  16. ...and 5 more sections

Key Result

Proposition 2.6

Let $(E_1,\varphi_1)$ and $(E_2,\varphi_2)$ be semistable $V$-twisted Higgs bundles on $X$. Then

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 36 more