Abstract:Let $f:X\to S$ be a smooth proper family of smooth projective varieties, and let $\sigma_{\mathrm{Dol}}:\,S \to M_{\mathrm{Dol}}(X/S)$ be the real analytic family of Higgs bundles obtained from an isomonodromic deformation via the relative non-abelian Hodge correspondence. We study the interaction between isomonodromic deformation and the natural $\mathbb C^*$-action on Dolbeault moduli spaces. For $\lambda\in S^1$, we prove that, on any complex analytic subvariety $U\subset S$, the rescaled family $\lambda\cdot\sigma_{\mathrm{Dol}}|_U$ is again isomonodromic if $\sigma_{\mathrm{Dol}}|_U$ is holomorphic. Conversely, we prove that $\sigma_{\mathrm{Dol}}|_U$ must be holomorphic if there exists $\lambda\in S^1\backslash\{\pm 1\}$ such that $\lambda\cdot\sigma_{\mathrm{Dol}}|_U$ is isomonodromic. The proof is based on the study of real analytic deformations of Higgs bundles and the variation of harmonic metrics.
As an application, we give a simplified proof of a local characterization of Simpson's non-abelian Noether--Lefschetz locus firstly proved in \cite[Theorem 1.2]{HSJZ}. Namely, if the initial local system underlies a polarized complex variation of Hodge structures, then the non-abelian Noether--Lefschetz locus is precisely the maximal complex analytic subvariety of $S$ on which the real analytic section $\sigma_{\mathrm{Dol}}$ becomes holomorphic. This gives an affirmative answer to a question of Esnault and Kerz.
From: Tianzhi Hu [view email]
[v1]
Wed, 17 Jun 2026 07:29:55 UTC (26 KB)
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