Quadratic one-forms on logarithmic Higgs moduli

Abstract

Let $C$ be a compact Riemann surface of genus at least two, and let $G$ be a connected complex reductive group. We study logarithmic \(G\)-Higgs bundles on a pointed curve $(C,D)$ whose residues are nilpotent. We prove that every homogeneous invariant polynomial in the Higgs field loses its leading pole term. Equivalently, the nilpotent-residue logarithmic Hitchin map takes values in a smaller meromorphic Hitchin base. In degree two this gives a logarithmic quadratic one-form over pointed Teichmuller space. We relate this one-form to the variation of the energy for tame nilpotent harmonic bundles, under a positive-decay assumption near the punctures.