Holomorphic isomonodromic deformations of Higgs bundles: absolute lifting, spectral flatness, and nilpotent rigidity

Abstract

Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $σ_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section $σ_{\mathrm{Dol}}:S\to M_{\mathrm{Dol}}(X/S)$, whose value at each $s\in S$ is the Higgs bundle corresponding to the flat bundle $σ_{\mathrm{dR}}(s)$. Unlike $σ_{\mathrm{dR}}$, the section $σ_{\mathrm{Dol}}$ 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.