On birational Torelli theorems

TL;DR

This paper establishes new birational Torelli theorems linking the geometry of moduli spaces of principal bundles and parabolic bundles to the isomorphism class of the underlying curves, for genus at least 3 and 4 respectively.

Contribution

It proves the first 2-birational and 3-birational Torelli theorems connecting moduli space birationality to curve isomorphism.

Findings

2-birational Torelli theorem for principal G-bundles

3-birational Torelli theorem for symplectic parabolic bundles

Curves are determined by the birational class of their moduli spaces

Abstract

Let $G$ be a simple simply-connected connected linear algebraic group over $\mathbb{C}$. We proved a $2$-birational Torelli theorem for the moduli space of semistable principal $G$-bundles over a smooth curve of genus $\geq 3$, which says that if two such moduli spaces are $2$-birational then the curves are isomorphic. We also proved a $3$-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus $\geq 4$.