Non-Abelian Kodaira-Spencer Map and non-existence of holomorphic isomonodromic deformation of Higgs bundles over Teichm\"uller spaces

Abstract

We define the isomonodromic deformation of a Higgs bundle on a compact Riemann surface via the Hitchin--Simpson correspondence and the isomonodromic deformation of the associated local system. This construction yields a real-analytic section of the relative Dolbeault moduli space and hence a real-analytic foliation, generalizing the Betti foliation arising from the Betti map in the study of abelian schemes. We give cohomological expressions for the holomorphic and anti-holomorphic derivatives of the isomonodromic deformation and use the latter to extend the classical non-abelian Kodaira--Spencer map. We prove that if the isomonodromic deformation of a graded Higgs bundle is non-holomorphic, then the deformed Higgs field is non-nilpotent. We also give a short new proof of the non-existence of holomorphic isomonodromic deformations for generic Higgs bundles over Teichm\"uller space $\mathcal T_g$, previously established in \cite{biswas}. This shows that global holomorphicity imposes strong restrictions on the initial Higgs bundle. Motivated by this observation, we prove, under suitable numerical conditions, that non-nilpotent or non-unitary Higgs bundles have non-holomorphic isomonodromic deformations over $\mathcal T_g$. These results may be viewed as analogues of the Landesman--Litt finite-image theorem for MCG-finite representations \cite{LL}. This paper synthesizes and refines our two earlier preprints \cite{HSZ,HSZII} (arXiv:2511.14272 and arXiv:2512.15478), and makes further progress on the non-existence problem for holomorphic isomonodromic deformations of higher rank Higgs bundles over Teichm\"uller spaces.