Isomonodromic deformations, $\mathbb C^*$-actions, and characterization of non-abelian Noether-Lefschetz loci on Dolbeault moduli spaces

Abstract: Let $f:X\to S$ be a smooth proper family of smooth projective varieties, and let $σ{\mathrm{Dol}}:\,S \to M{\mathrm{Dol}}(X/S)$ 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 $λ\in S^1$, we prove that, on any complex analytic subvariety $U\subset S$, the rescaled family $λ\cdotσ{\mathrm{Dol}}|_U$ is again isomonodromic if $σ{\mathrm{Dol}}|U$ is holomorphic. Conversely, we prove that $σ{\mathrm{Dol}}|U$ must be holomorphic if there exists $λ\in S^1\backslash{\pm 1}$ such that $λ\cdotσ{\mathrm{Dol}}|U$ 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 $σ{\mathrm{Dol}}$ becomes holomorphic. This gives an affirmative answer to a question of Esnault and Kerz.