[Paper Review] Spectral Covers

Ron Donagi|arXiv (Cornell University)|May 7, 1995

Advanced Algebra and GeometryMathematics67 citations

TL;DR

This paper surveys spectral covers associated with G-principal Higgs bundles, introducing a W-Galois cameral cover $ tilde{S}$ of a variety $S$ and analyzing the decomposition of the Picard group of $ tilde{S}$ into Prym varieties. It identifies a distinguished Prym component as a moduli space for Higgs bundles, extending Hitchin's abelianization program via representation-theoretic spectral data.

ABSTRACT

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

Motivation & Objective

• To unify and survey the theory of spectral covers in the context of G-principal Higgs bundles.
• To clarify the geometric and cohomological structure of spectral covers arising from representations of the reductive group G.
• To interpret the decomposition of the Picard group of the cameral cover into Prym varieties as a generalization of known Prym identities.
• To identify a distinguished Prym component as a parameter space for Higgs bundles, in line with Hitchin's abelianization program.
• To establish a framework for understanding Higgs bundle moduli through equivariant sheaves on cameral covers.

Proposed method

• Construct a cameral cover $ tilde{S} o S$ as a W-Galois cover, where W is the Weyl group of G.
• Associate to each G-Higgs bundle a W-equivariant sheaf on $ tilde{S}$, twisted by canonical line bundles and shifts.
• Use representation theory to define alternative spectral covers via different representations or weights of G.
• Analyze the Picard group of $ tilde{S}$ by decomposing it into Prym varieties associated to subgroups of W.
• Apply the abelianization program to interpret the distinguished Prym component as a moduli space for Higgs bundles.
• Utilize equivariance and twist data to relate sheaf-theoretic data on $ tilde{S}$ to Higgs bundle structures on S.

Experimental results

Research questions

• RQ1 How does the cameral cover $ tilde{S}$ of a variety S encode the structure of a G-principal Higgs bundle on S?
• RQ2 What is the role of the Weyl group W in constructing the spectral cover and its associated sheaf?
• RQ3 How does the Picard group of the cameral cover decompose into Prym varieties, and what is the significance of this decomposition?
• RQ4 Which Prym component of the spectral cover corresponds to the moduli space of Higgs bundles, and how is it characterized?
• RQ5 In what way does the spectral cover construction realize Hitchin's abelianization program for Higgs bundles?

Key findings

• The cameral cover $ tilde{S}$ is a W-Galois cover of S, constructed from the Weyl group of G, and serves as the geometric base for spectral data.
• A G-Higgs bundle on S corresponds to a W-equivariant sheaf on $ tilde{S}$, twisted by canonical line bundles and shifts.
• The Picard group of $ tilde{S}$ decomposes into Prym varieties associated to subgroups of W, generalizing classical Prym identities.
• A distinguished Prym component of the spectral cover's Picard group parameterizes the moduli space of Higgs bundles on S.
• The construction realizes Hitchin's abelianization program by identifying a moduli space for Higgs bundles via the geometry of the spectral cover and its Prym decomposition.