Rank-2 wobbly bundles from special divisors on spectral curves

TL;DR

This paper characterizes rank-2 wobbly bundles on a Riemann surface using spectral curves and divisors, providing conditions for their existence and exploring the geometry of their moduli space.

Contribution

It introduces a criterion for identifying wobbly bundles as direct images of line bundles on spectral curves, linking their properties to divisors of quadratic differentials.

Findings

Wobbly bundles are twists of direct images of line bundles on spectral curves.

Necessary and sufficient conditions for a bundle to be wobbly are established.

Singularities in the wobbly locus relate to Brill-Noether loci of spectral curves.

Abstract

We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C} \overset{\pi}{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $\pi_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ where the norm of $\tilde{D}$ is a summand of the divisor of a quadratic differential on $C$. We sketch the proof of the necessary condition statement, namely all rank-2 wobbly bundles can be characterised as such, and discuss how certain singularities of the wobbly locus arise from the Brill-Noether loci of spectral curves.