Fourier–Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes

Abstract

Given a vector bundle E on a smooth projective curve or surface X carrying the structure of a V-twisted Hitchin pair for some vector bundle V, we observe that the associated tautological bundle E[n] on the punctual Hilbert scheme of points X[n] has an induced structure of a ((V∨)[n])∨-twisted Hitchin pair, where (V∨)[n] is a vector bundle on X[n] constructed using the dual V∨ of V. In particular, a Higgs bundle on X induces a logarithmic Higgs bundle on the Hilbert scheme X[n]. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs.