Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stratifying moduli spaces of Higgs bundles and the Hitchin morphism

Aryaman Patel, Dario Weissmann

TL;DR

This work analyzes how slope-stability for reflexive twisted sheaves behaves under reflexive pullback along finite covers, introducing genuine ramification in codimension 1 as a key criterion. It proves the existence of good and split quasi-étale covers that reduce stability questions to checking stability on a single cover, and uses Galois symmetry to understand decomposition into rank-1 summands, yielding a stratification of the moduli of slope-stable F-twisted Higgs bundles by decomposition type. It then constructs and analyzes the Hitchin morphism on the smallest semistable stratum, determining its image and applying these results to Dolbeault moduli spaces on hyperelliptic and abelian varieties; in characteristic zero this confirms Chen-Ngô-type predictions for these cases. The paper provides a cohesive framework for descent, stability, and splitting phenomena in twisted Higgs theory, with concrete consequences for Hitchin bases and spectral data.

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-étale cover. Fixing the rank, there is one quasi-étale 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-Ngô's conjecture holds for hyperelliptic varieties in characteristic $0$.

Stratifying moduli spaces of Higgs bundles and the Hitchin morphism

TL;DR

This work analyzes how slope-stability for reflexive twisted sheaves behaves under reflexive pullback along finite covers, introducing genuine ramification in codimension 1 as a key criterion. It proves the existence of good and split quasi-étale covers that reduce stability questions to checking stability on a single cover, and uses Galois symmetry to understand decomposition into rank-1 summands, yielding a stratification of the moduli of slope-stable F-twisted Higgs bundles by decomposition type. It then constructs and analyzes the Hitchin morphism on the smallest semistable stratum, determining its image and applying these results to Dolbeault moduli spaces on hyperelliptic and abelian varieties; in characteristic zero this confirms Chen-Ngô-type predictions for these cases. The paper provides a cohesive framework for descent, stability, and splitting phenomena in twisted Higgs theory, with concrete consequences for Hitchin bases and spectral data.

Abstract

We study the behavior of slope-stability of reflexive twisted sheaves over a normal projective variety under pullback along a cover. Slope-stability is always preserved if the cover does not factor via a quasi-étale cover. Fixing the rank, there is one quasi-étale 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 is smooth. This allows us to determine the image of the Hitchin morphism from the Dolbeault moduli space when is a hyperelliptic or abelian variety in characteristic . In particular, we show that Chen-Ngô's conjecture holds for hyperelliptic varieties in characteristic .
Paper Structure (13 sections, 42 theorems, 45 equations)

This paper contains 13 sections, 42 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Reflexive twisted sheaves
  3. Genuinely ramified morphisms and stability
  4. Étale fundamental group of the smooth locus
  5. Covers genuinely ramified in codimension 1
  6. Genuinely ramified in codimension 1 and stability: the Galois case
  7. Genuinely ramified in codimension 1 and stability: the general case
  8. Functorial notions of stability
  9. Decomposition strata
  10. Moduli
  11. The split cover and strata
  12. The image of the semistable 1-stratum under the Hitchin morphism
  13. Specializing to classical Higgs bundles

Key Result

Theorem 1.1

Let $\pi:Y\to X$ be a finite separable morphism of normal projective varieties that does not factor through a non-trivial finite étale morphism, and let $F$ be a vector bundle on $X$. Then an $F$-twisted vector bundle $(\mathcal{V},\theta)$ on $X$ is slope-stable iff $(\pi^{\ast}\mathcal{V},\pi^{\as

Theorems & Definitions (100)

  • Theorem 1.1: Theorem \ref{['thm:bdp analogue stable']}
  • Theorem 1.2: Theorem \ref{['thm:good cover']}, char $0$
  • Theorem 1.3: Lemma \ref{['H-split']} + Theorem \ref{['1-stratum image']}, char $0$
  • Theorem 1.4: Corollary \ref{['hyperelliptic']}
  • Lemma 2.1: reflexive-hartshorne
  • Lemma 2.2: reflexive-hartshorne
  • Definition 2.3: Category of reflexive $F$-twisted sheaves
  • Remark 2.4
  • Definition 2.5: Cover
  • Lemma 2.6
  • ...and 90 more