Isomonodromic deformations of Higgs bundles and characterization of the non-abelian Noether--Lefschetz locus

Abstract

Let $f:X\to S$ be a smooth proper family of smooth projective varieties. An irreducible complex local system on a fiber admits an isomonodromic deformation, hence determines a holomorphic section of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence produces a real-analytic section $\sigma_{Dol}:S\to M_{Dol}(X/S)$ of the relative Dolbeault moduli space. In this paper, we investigate when this real-analytic section is holomorphic. The first approach uses the first-order infinitesimal deformation: we prove a Cauchy--Riemann type criterion showing that holomorphicity in a tangent direction of $S$ is measured by the composition of the Kodaira--Spencer map with the non-abelian Higgs field. The second approach involves higher-order derivatives: after restricting $\sigma_{Dol}$ to infinitesimal thickenings of the reference point in $S$, we introduce obstruction classes measuring the failure of holomorphicity and relate them to the Taylor expansion of the harmonic metric. We apply these criteria to three problems. First, we study the interaction between the $\mathbb C^*$-action on Higgs bundles and isomonodromic deformations.Second, for an initial polarized complex variation of Hodge structures, we consider the associated non-abelian Noether--Lefschetz locus. We prove that this locus is precisely the maximal complex analytic subvariety of $S$ on which the real-analytic isomonodromic deformation $\sigma_{Dol}$ becomes holomorphic. Both the first-order and higher-order methods yield proofs of this characterization. Lastly, we prove that if the initial Higgs bundle is generically regular nilpotent and the isomonodromic deformation is holomorphic, then every member of the family is represented by a nilpotent Higgs bundle.