Non-existence of holomorphic isomonodromic deformation of a Higgs bundle

TL;DR

This work proves the non-existence of holomorphic isomonodromic deformation for generic $\mathrm{SL}(n,\mathbb C)$-Higgs bundles and for certain low-rank non-unitary Higgs bundles over the Teichmüller space. Building on the cohomological interpretation of anti-holomorphic derivatives from HSZ and a holomorphic-deformation criterion, the authors reduce the problem to analyzing the map $\theta_*$ through the kernel of $\mathrm{ad}(\theta)$ and, in many cases, to spectral-curve data via the BNR correspondence. They establish injectivity or non-vanishing of $\theta_*$ in generic and rank-specific scenarios (rank 2 and 3), including nilpotent and graded/non-unitary cases, using cohomological, duality, and degree arguments. The results yield a simpler, more uniform proof framework compared to previous approaches (e.g., Biswas) and clarify obstructions to holomorphic isomonodromic deformations in these moduli spaces.

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 Teichmüller 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 Teichmüller space $\mathcal T_g$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.1.1
• proof