Non-existence of holomorphic isomonodromic deformation of a Higgs bundle

We use the cohomological interpretation of anti-holomorphic derivatives of the isomonodromic deformation of a Higgs bundle, as established in our previous work \cite{HSZ}, to provide a short new proof of the non-existence of holomorphic isomonodromic deformation of a generic $\SL$-Higgs bundle and of any non-unitary rank 2 Higgs bundle over the Teichm\"uller space $\mathcal T_g$, which were previously proved in \cite{biswas}. We also prove the non-existence of holomorphic isomonodromic deformation of any non-unitary rank 3 Higgs bundle over the Teichm\"uller space $\mathcal T_g$.