A generalized spectral correspondence

TL;DR

This paper establishes a categorical spectral correspondence between sheaves on algebraic curves and twisted pairs, applying it to elliptic curves and constructing examples of special bundles, with a proven conjecture for specific spectral covers.

Contribution

It introduces a generalized spectral correspondence framework and proves a conjecture for certain spectral covers using Galois theory, expanding understanding of sheaves and twisted pairs.

Findings

Established a categorical spectral correspondence between sheaves and twisted pairs.

Constructed examples of cyclic pairs and co-Higgs bundles over .

Proved a conjecture for spectral covers of  using Galois-theoretic classification.

Abstract

We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of cyclic pairs and co-Higgs bundles over $\mathbb{P}^1$. By appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$ through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt.