Kodaira-Spencer Map on the Hitchin-Simpson Correspondence
Tianzhi Hu, Mai Shi, Ruiran Sun, Kang Zuo
TL;DR
This work studies isomonodromic deformation of Higgs bundles through the Hitchin-Simpson correspondence, producing a real-analytic section of the relative Dolbeault moduli and a corresponding foliation. It derives explicit holomorphic and anti-holomorphic Kodaira-Spencer maps, extending the non-abelian KS framework via anti-holomorphic data and linking unitary cases to holomorphic deformations. For graded Higgs bundles, it proves a non-nilpotency result when the anti-holomorphic KS component is nonzero and explores how nilpotent behavior relates to the Hitchin fibration and Torelli-type phenomena, including conjectures on nilpotent loci. The paper also clarifies the relation between the Betti map and isomonodromic deformations, situating the results within non-abelian Hodge theory for families and the geometry of moduli spaces.
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.