On the nilpotent residue non-abelian Hodge correspondence for higher-dimensional quasiprojective varieties

TL;DR

This paper establishes that a known bijection between certain moduli spaces of nilpotent Higgs bundles and connections on higher-dimensional quasiprojective varieties is actually a homeomorphism, strengthening the understanding of the non-abelian Hodge correspondence.

Contribution

It proves that the map between the moduli spaces of nilpotent Higgs bundles and connections is a homeomorphism, extending previous bijection results to a topological equivalence.

Findings

The bijection is a homeomorphism.

Strengthens the non-abelian Hodge correspondence.

Applies to higher-dimensional quasiprojective varieties.

Abstract

In arXiv:2408.16441, the authors proved that on a projective log smooth variety $(\bar{X}, D)$ there is a continuous bijection between the moduli space $M^{\mathrm{nilp}}_{\mathrm{Dol}}(\bar{X}, D)$ of logarithmic Higgs bundles with nilpotent residues and the moduli space $M^{\mathrm{nilp}}_{\mathrm{DR}}(\bar{X}, D)$ of logarithmic connections with nilpotent residues. In this note, we argue that the map is a homeomorphism.