Stratifying moduli spaces of Higgs bundles and the Hitchin morphism

TL;DR

This paper investigates the structure of moduli spaces of Higgs bundles via stratification induced by quasi-étale covers, and applies this to analyze the Hitchin morphism's image, confirming Chen-Ngô's conjecture for hyperelliptic varieties in characteristic zero.

Contribution

It introduces a stratification of the moduli space based on slope-stability preservation under covers and determines the Hitchin morphism's image for specific varieties.

Findings

Stratification of moduli space via quasi-étale covers.

Identification of the Hitchin morphism's image for hyperelliptic and abelian varieties.

Confirmation of Chen-Ngô's conjecture for hyperelliptic varieties in characteristic zero.

Abstract

We study the behavior of slope-stability of reflexive twisted sheaves over a normal projective variety $X$ under pullback along a cover. Slope-stability is always preserved if the cover does not factor via a quasi-\'etale cover. Fixing the rank, there is one quasi-\'etale cover that checks whether a twisted sheaf remains slope-stable on all Galois covers, yielding a stratification of the moduli space of slope-stable Higgs-bundles. As an application, we determine the image of the Hitchin morphism restricted to the smallest closed stratum of the Dolbeault moduli space when $X$ is smooth. This allows us to determine the image of the Hitchin morphism from the Dolbeault moduli space when $X$ is a hyperelliptic or abelian variety in characteristic $p\ge0$. In particular, we show that Chen-Ng\^o's conjecture holds for hyperelliptic varieties in characteristic $0$.