A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

TL;DR

This paper proves a Torelli theorem for the moduli space of rank two parabolic vector bundles over an elliptic curve, establishing a link between the moduli space and the underlying curve.

Contribution

It establishes a Torelli-type result for moduli spaces of parabolic bundles over elliptic curves, a novel extension of classical Torelli theorems.

Findings

The moduli space uniquely determines the elliptic curve and marked points.

The Torelli theorem holds for semistable parabolic bundles with specific weights.

The result extends classical Torelli theorems to parabolic bundle moduli spaces.

Abstract

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big(\frac{1}{2}, \dots, \frac{1}{2}\big)$.