Rank 3 curve semistable Higgs bundles: the nilpotent case
Abstract
A Higgs bundle on a smooth complex projective variety is curve semistable if its pullback to every smooth projective curve is semistable. A 2008 conjecture asserts that this is equivalent to slope semistability for some polarization plus vanishing discriminant. One direction is known, the converse holds in rank two, and Huang recently gave a rank-four counterexample. We prove the conjecture for rank three nilpotent Higgs bundles in any dimension. We treat separately the two possible cases: if the Higgs field ϕ is nilpotent of order two, there is a divisorial component of a Higgs Grassmannian where nefness of the normalized tautological class forces the discriminant to vanish; otherwise, ϕ is regular nilpotent, and then the canonical kernel flag, commutativity and Bogomolov's lemma on line subbundles of the cotangent bundle complete the proof.
Related
- 𝔸1-connectedness of moduli of semistable bundles and symplectic bundles on a curve
- Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity
- Counting Irreducible Components of the Semistable Global Nilpotent Cone
- Geometry of a semistable degeneration of moduli of vector bundles on curves
- Collisions of Nilpotent Higgs Fields
BibTeX
@misc{bruzzo2026rank,
title = {{Rank 3 curve semistable Higgs bundles: the nilpotent case}},
author = {Ugo Bruzzo and Beatriz Graña Otero},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32900},
eprint = {2610.32900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32900},
}
RIS
TY - UNPB
TI - Rank 3 curve semistable Higgs bundles: the nilpotent case
AU - Ugo Bruzzo
AU - Beatriz Graña Otero
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32900
UR - https://arxiv.org/abs/2610.32900
AB - A Higgs bundle on a smooth complex projective variety is curve semistable if its pullback to every smooth projective curve is semistable. A 2008 conjecture asserts that this is equivalent to slope semistability for some polarization plus vanishing discriminant. One direction is known, the converse holds in rank two, and Huang recently gave a rank-four counterexample. We prove the conjecture for rank three nilpotent Higgs bundles in any dimension. We treat separately the two possible cases: if the Higgs field ϕ is nilpotent of order two, there is a divisorial component of a Higgs Grassmannian where nefness of the normalized tautological class forces the discriminant to vanish; otherwise, ϕ is regular nilpotent, and then the canonical kernel flag, commutativity and Bogomolov's lemma on line subbundles of the cotangent bundle complete the proof.
ER -