Torelli theorem for the moduli space of framed bundles

TL;DR

This paper proves a Torelli theorem for the moduli space of au-semistable framed bundles over a smooth complex projective curve, showing that the moduli space uniquely determines the curve and the rank for small au.

Contribution

It establishes a Torelli theorem for the moduli space of framed bundles, linking the geometry of the moduli space to the underlying curve and bundle rank.

Findings

The moduli space M^ au uniquely determines the curve (X,x) for small au.

The integer rank r of the bundles is recoverable from M^ au.

The Torelli theorem holds for au > 0 sufficiently small.

Abstract

Let X be an irreducible smooth complex projective curve of genus g>2, and let x be a fixed point. A framed bundle is a pair (E,\phi), where E is a vector bundle over X, of rank r and degree d, and \phi:E_x\to C^r is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter \tau>0, which gives rise to the moduli space of \tau-semistable framed bundles M^\tau. We prove a Torelli theorem for M^\tau, for \tau>0 small enough, meaning, the isomorphism class of the one-pointed curve (X,x), and also the integer r, are uniquely determined by the isomorphism class of the variety M^\tau.