Kodaira-Spencer Map on the Hitchin-Simpson Correspondence

TL;DR

This paper explores the isomonodromic deformation of Higgs bundles on Riemann surfaces via the Hitchin-Simpson correspondence, extending classical Kodaira-Spencer theory and analyzing the nature of deformations.

Contribution

It introduces a new framework for isomonodromic deformations in Higgs bundles, extending the Kodaira-Spencer map and analyzing the holomorphic properties of these deformations.

Findings

Defined a real analytic foliation on the moduli space.

Extended the non-abelian Kodaira-Spencer map using anti-holomorphic derivatives.

Proved non-holomorphic deformations imply non-nilpotent Higgs fields.

Abstract

We define the isomonodromic deformation of a Higgs bundle over a compact Riemann surface via the Hitchin-Simpson correspondence and the isomonodromic deformation of a local system. This deformation defines a real analytic section of the relative Dolbeault moduli space, yielding a real analytic foliation on this moduli. This foliation generalizes the Betti foliation defined by the Betti map in the study of abelian schemes. We provide a precise form for the holomorphic and anti-holomorphic derivatives of the isomonodromic deformation of a Higgs bundle. Subsequently, we extend the classical non-abelian Kodaira-Spencer map using the anti-holomorphic derivative. Additionally, we prove that if the isomonodromic deformation of a graded Higgs bundle is not holomorphic, then the isomonodromically deformed Higgs field is non-nilpotent.