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PreprintCC BY 4.0arXiv

Local obstructions to the surjectivity of spectral morphisms

  • Siqi He
  • Jie Liu
  • Siqing Zhang
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Abstract

Chen and Ngo showed that the Hitchin morphism for Higgs bundles on a projective manifold X factors through a closed subscheme of the Hitchin base called the spectral base, and the induced map is called the spectral morphism. In this paper, we investigate local Cohen-Macaulay obstructions to the surjectivity of spectral morphisms via a refined spectral correspondence. We show that every affine normal variety with a local ring that admits no rank one maximal Cohen-Macaulay modules can be geometrically realized as a local obstruction to the surjectivity of spectral morphisms, and that these local obstructions can be compactified via the Kawamata covering trick. As an application of these constructions, we provide counterexamples to the Chen-Ngo conjecture on the surjectivity of spectral morphisms for GLr-Higgs bundles in dimensions n≥ 9. Moreover, we construct counterexamples to the Chen-Ngo conjecture for SL2-Higgs bundles in every dimension n≥ 2 and for semistable GL3-Higgs bundles on a smooth projective surface.

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Cite

BibTeXRIS
BibTeX
@misc{he2026local,
  title = {{Local obstructions to the surjectivity of spectral morphisms}},
  author = {Siqi He and Jie Liu and Siqing Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17139},
  eprint = {2609.17139},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17139},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Local obstructions to the surjectivity of spectral morphisms
AU  - Siqi He
AU  - Jie Liu
AU  - Siqing Zhang
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17139
UR  - https://arxiv.org/abs/2609.17139
AB  - Chen and Ngo showed that the Hitchin morphism for Higgs bundles on a projective manifold X factors through a closed subscheme of the Hitchin base called the spectral base, and the induced map is called the spectral morphism. In this paper, we investigate local Cohen-Macaulay obstructions to the surjectivity of spectral morphisms via a refined spectral correspondence. We show that every affine normal variety with a local ring that admits no rank one maximal Cohen-Macaulay modules can be geometrically realized as a local obstruction to the surjectivity of spectral morphisms, and that these local obstructions can be compactified via the Kawamata covering trick. As an application of these constructions, we provide counterexamples to the Chen-Ngo conjecture on the surjectivity of spectral morphisms for GL_r-Higgs bundles in dimensions n≥ 9. Moreover, we construct counterexamples to the Chen-Ngo conjecture for SL_2-Higgs bundles in every dimension n≥ 2 and for semistable GL_3-Higgs bundles on a smooth projective surface.
ER  -