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 quadratic one-forms associated to logarithmic $G$-Higgs bundles on a pointed curve $(C,D)$ with nilpotent residues. We use the elementary pole cancellation for invariant polynomials, where nilpotency of the residue removes the leading pole term. In degree two this places $B(\hat{I}¦,\hat{I}¦)$ in the cotangent space of pointed Teichmuller space, and hence gives a logarithmic quadratic one-form. We relate this one-form to the variation of the energy for tame nilpotent harmonic bundles. The energy formula is proved under a positive decay assumption near the punctures. We also write the corresponding compact formulae for classical groups and standard real forms.