Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity
Abstract
Let f:X→ S be a smooth projective family, and fix a semisimple flat bundle on a fiber X0. Its isomonodromic deformation determines a holomorphic section σdR:S→ MdR(X/S) of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section σDol:S→ MDol(X/S), whose value at each s∈ S is the Higgs bundle corresponding to the flat bundle σdR(s). Unlike σdR, the section σDol is in general only real analytic. We study the geometric consequences of its holomorphicity along a complex analytic subvariety. First, we prove an absolute lifting theorem: the relative isomonodromic Higgs bundle admits an absolute Higgs lift, and the fiberwise harmonic metrics can be modified to solve the Hitchin--Simpson equation on the total space. Second, we prove that the spectral one-form on every resolved irreducible component of the relative spectral scheme is Gauss--Manin flat. As an application, we prove the nilpotent rigidity conjecture of Hu--Sun--Yang--Zuo: if the initial Higgs field is nilpotent, then the Higgs field remains nilpotent along the entire holomorphic isomonodromic locus.
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BibTeX
@misc{hu2026holomorphic,
title = {{Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity}},
author = {Tianzhi Hu and Mai Shi and Kang Zuo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12416},
eprint = {2609.12416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12416},
}
RIS
TY - UNPB
TI - Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity
AU - Tianzhi Hu
AU - Mai Shi
AU - Kang Zuo
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.12416
UR - https://arxiv.org/abs/2609.12416
AB - Let f:X→ S be a smooth projective family, and fix a semisimple flat bundle on a fiber X_0. Its isomonodromic deformation determines a holomorphic section σ_dR:S→ M_dR(X/S) of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section σ_Dol:S→ M_Dol(X/S), whose value at each s∈ S is the Higgs bundle corresponding to the flat bundle σ_dR(s). Unlike σ_dR, the section σ_Dol is in general only real analytic. We study the geometric consequences of its holomorphicity along a complex analytic subvariety. First, we prove an absolute lifting theorem: the relative isomonodromic Higgs bundle admits an absolute Higgs lift, and the fiberwise harmonic metrics can be modified to solve the Hitchin--Simpson equation on the total space. Second, we prove that the spectral one-form on every resolved irreducible component of the relative spectral scheme is Gauss--Manin flat. As an application, we prove the nilpotent rigidity conjecture of Hu--Sun--Yang--Zuo: if the initial Higgs field is nilpotent, then the Higgs field remains nilpotent along the entire holomorphic isomonodromic locus.
ER -