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PreprintarXiv licencearXiv

Simpson's closedness conjecture in arbitrary rank

  • Tianzhi Hu
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Abstract

For a compact Riemann surface X, Simpson associated to each stable graded Higgs bundle (E,θ) a locus W[(E,θ)]1 ⊂ MdR(X,n), consisting of flat bundles that admit a Simpson filtration whose associated graded Higgs bundle is (E,θ). He conjectured that W[(E,θ)]1 is Zariski closed in MdR(X,n). We prove this conjecture in arbitrary rank. The proof is based on the Quillen geometry of the determinant-of-cohomology line bundle over the moduli of holomorphic bundles. To any holomorphic family of flat bundles, we associate a determinant-of-cohomology line bundle equipped with a canonical holomorphic determinant connection, and derive explicit formulas for its curvature. For a family of flat bundles in W[(E,θ)]1, we then prove that the determinant connection form is exact. This exactness forces the associated determinant frame to extend as a nowhere-vanishing frame across any one-parameter degeneration, which controls the limiting filtration and yields the desired closedness.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026simpsons,
  title = {{Simpson's closedness conjecture in arbitrary rank}},
  author = {Tianzhi Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03542},
  eprint = {2610.03542},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03542},
}
RIS
TY  - UNPB
TI  - Simpson's closedness conjecture in arbitrary rank
AU  - Tianzhi Hu
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.03542
UR  - https://arxiv.org/abs/2610.03542
AB  - For a compact Riemann surface X, Simpson associated to each stable graded Higgs bundle (E,θ) a locus W_[(E,θ)]^1 ⊂ M_dR(X,n), consisting of flat bundles that admit a Simpson filtration whose associated graded Higgs bundle is (E,θ). He conjectured that W_[(E,θ)]^1 is Zariski closed in M_dR(X,n). We prove this conjecture in arbitrary rank. The proof is based on the Quillen geometry of the determinant-of-cohomology line bundle over the moduli of holomorphic bundles. To any holomorphic family of flat bundles, we associate a determinant-of-cohomology line bundle equipped with a canonical holomorphic determinant connection, and derive explicit formulas for its curvature. For a family of flat bundles in W_[(E,θ)]^1, we then prove that the determinant connection form is exact. This exactness forces the associated determinant frame to extend as a nowhere-vanishing frame across any one-parameter degeneration, which controls the limiting filtration and yields the desired closedness.
ER  -