Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles

TL;DR

This paper establishes Torelli theorems for moduli spaces of semistable parabolic Higgs bundles and parabolic bundles on algebraic curves, linking their geometric structures to the underlying curves using integrable systems.

Contribution

It proves Torelli theorems for these moduli spaces under generic weights, extending previous results to parabolic Higgs bundles and higher rank bundles.

Findings

Torelli theorem for parabolic Higgs bundles over smooth curves

Torelli theorem for parabolic bundles of rank at least two when genus ≥ 2

Application of integrable systems methods in algebraic geometry

Abstract

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.