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

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 ${\sigma }_{\mathrm{d}\mathrm{R}}:S\to M_{\mathrm{d}\mathrm{R}}(X\mathrm{/}S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a section ${\sigma }_{\mathrm{D}\mathrm{o}\mathrm{l}}:S\to M_{\mathrm{D}\mathrm{o}\mathrm{l}}(X\mathrm{/}S)$, whose value at each $s\in S$ is the Higgs bundle corresponding to the flat bundle ${\sigma }_{\mathrm{d}\mathrm{R}}(s)$. Unlike ${\sigma }_{\mathrm{d}\mathrm{R}}$, the section ${\sigma }_{\mathrm{D}\mathrm{o}\mathrm{l}}$ 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.