Rank-2 wobbly bundles from special divisors on spectral curves

TL;DR

The paper develops a spectral-curve framework to classify rank-2 wobbly bundles on a genus $g$ curve, linking semi-stable bundles to direct images $\pi_*\mathcal{L}$ from smooth spectral curves and using the Hitchin fibration to study wobbliness. It proves a key sufficient condition: if $\mathcal{L}=\pi^*L\otimes\mathcal{O}_{\tilde{C}}(\tilde{D})$ with $\tilde{D}$ being $Q$-special (i.e., $\mathrm{Nm}(\tilde{D})$ divides a quadratic differential), then the corresponding $E=\pi_*\mathcal{L}$ is wobbly; it also provides a route toward necessity via the Donagi–Pantev resolution of the direct-image map and analyzes the $SL_2$ case, including a decomposition of the wobbl y locus into divisors $\mathcal{W}_k$. The work further connects wobbliness to Brill-Noether data on spectral curves, showing how BN loci influence the dimension of nilpotent Higgs spaces and the singular structure of the wobbl y locus, with implications for the geometric Langlands program through the interplay with Theta divisors and Prym varieties.

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 $\tilde{C} \oversetπ{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $π_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ 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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 2.4