# Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes

## TL;DR

This paper explores how vector bundles with Hitchin pair structures on curves or surfaces induce related structures on punctual Hilbert schemes, extending stability and reconstruction results to these tautological Hitchin pairs.

## Contribution

It introduces a method to induce Hitchin pair structures on tautological bundles over Hilbert schemes, generalizing stability and reconstruction results.

## Key findings

- Tautological bundles inherit Hitchin pair structures.
- Higgs bundles induce logarithmic Higgs bundles on Hilbert schemes.
- Stability and reconstruction results extend to tautological Hitchin pairs.

## 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 $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ 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.