Non-existence of holomorphic isomonodromic deformation of a Higgs bundle

TL;DR

This paper proves that holomorphic isomonodromic deformations do not exist for generic and non-unitary Higgs bundles of ranks 2 and 3 over Teichmüller space, using a cohomological approach.

Contribution

It provides a new, concise proof of the non-existence of holomorphic isomonodromic deformations for certain Higgs bundles, extending previous results to rank 3.

Findings

Holomorphic isomonodromic deformations do not exist for generic rank 2 Higgs bundles.

Holomorphic isomonodromic deformations do not exist for non-unitary rank 3 Higgs bundles.

The proof uses a cohomological interpretation of anti-holomorphic derivatives of isomonodromic deformations.

Abstract

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 $\mathcal T_g$.