[Paper Review] The gerbe of Higgs bundles

Ron Donagi, Dennis Gaitsgory|ArXiv.org|May 13, 2000

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper establishes that the category of Higgs bundles with a fixed cameral cover ${\widetilde{X}}$ over a complex scheme $X$ forms a gerbe banded by the sheaf of abelian groups $T_{{\widetilde{X}}}$, whose class in $H^2(X, T_{{\widetilde{X}}})$ is determined by three components: a ramification twist, a Weyl group cohomology class $[N] \in H^2(W, T)$, and a third term for $SO(2n+1)$-type groups arising from non-primitive coroots. The isomorphism classes of such Higgs bundles form a torsor over $H^1(X, T_{{\widetilde{X}}})$, which generalizes the Prym variety.

ABSTRACT

The purpose of this work is to describe the (category of) Higgs bundles on a complex scheme X having a given cameral cover X~. We show that this category is a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we describe the class of this gerbe precisely. In particular, it follows that the set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors on X. This underlying group can be described as a generalized Prym variety, whose connected component is either an abelian variety or a degeneration thereof.

Motivation & Objective

• To classify Higgs bundles on a scheme $X$ over $\mathbb{C}$ with a fixed cameral cover ${\widetilde{X}}$.
• To show that the category of such Higgs bundles is a gerbe banded by the sheaf $T_{{\widetilde{X}}}$ of regular centralizers on ${\widetilde{X}}$.
• To precisely identify the class of this gerbe in $H^2(X, T_{{\widetilde{X}}})$, which governs the isomorphism classes of Higgs bundles.

Proposed method

• Define Higgs bundles as principal $G$-bundles equipped with a subbundle of regular centralizers in the adjoint bundle.
• Construct the cameral cover ${\widetilde{X}} \to X$ as a finite flat $W$-equivariant cover, generalizing spectral covers for $GL(n)$.
• Show that the category $\operatorname{Higgs}_{{\widetilde{X}}}(X)$ of Higgs bundles with fixed cameral cover is a gerbe banded by $T_{{\widetilde{X}}}$ via descent and torsor structure.
• Identify the gerbe class as a sum of three components: a ramification twist, a Weyl group cohomology class $[N] \in H^2(W, T)$, and a coroot-related term for $SO(2n+1)$.
• Use the universal example $\overline{G/N}$ to analyze the structure of the gerbe and the group-scheme of centralizers.
• Apply the results to elliptic fibrations, showing that generic semistable $G$-bundles on fibers are regular and admit unique regularization.

Experimental results

Research questions

• RQ1 How can Higgs bundles with a fixed cameral cover be classified in terms of abelian data?
• RQ2 What is the precise class in $H^2(X, T_{{\widetilde{X}}})$ that classifies the gerbe of Higgs bundles over a fixed cameral cover?
• RQ3 Why does the gerbe class include contributions from ramification, Weyl group cohomology $H^2(W, T)$, and non-primitive coroots?
• RQ4 How does the structure of the gerbe relate to generalized Prym varieties in the case of elliptic fibrations?
• RQ5 What is the role of the universal space $\overline{G/N}$ in understanding the global structure of the gerbe?

Key findings

• The category of Higgs bundles with a fixed cameral cover ${\widetilde{X}}$ is a gerbe banded by the sheaf $T_{{\widetilde{X}}}$, meaning isomorphism classes form a torsor over $H^1(X, T_{{\widetilde{X}}})$.
• The class of this gerbe in $H^2(X, T_{{\widetilde{X}}})$ is the sum of three components: a ramification twist along the branch locus of ${\widetilde{X}} \to X$, a Weyl group cohomology class $[N] \in H^2(W, T)$, and a third term for $SO(2n+1)$-type groups arising from non-primitive coroots.
• The group $H^1(X, T_{{\widetilde{X}}})$ is a generalized Prym variety, whose connected component is either an abelian variety or a degenerate abelian variety.
• For $G = GL(n)$, $PGL(n)$, $SL(2n+1)$, and $SO(n)$, the Weyl group cohomology class $[N]$ vanishes, but it does not vanish for $SL(2n)$, indicating a nontrivial obstruction.
• In the case of elliptic fibrations, generic semistable $G$-bundles on fibers are regular, and their global structure is governed by the same gerbe-theoretic framework.
• The construction of the gerbe is compatible with base change and applies to both ramified and unramified cameral covers, with the universal example $\overline{G/N}$ providing a global model for the moduli space.