Torelli theorem for the moduli spaces of pairs

TL;DR

This paper proves a Torelli theorem for moduli spaces of -stable pairs over algebraic curves, showing their third cohomology groups are isomorphic to the curve's first cohomology, revealing deep geometric connections.

Contribution

It establishes that the third cohomology of moduli spaces of -stable pairs is a polarised pure Hodge structure isomorphic to H^1(X), extending Torelli theorems to these moduli spaces.

Findings

Third cohomology groups are polarised pure Hodge structures.

Isomorphism between moduli space cohomology and H^1(X).

Torelli theorems for moduli spaces of pairs and bundles.

Abstract

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,\phi) over X consists of an algebraic vector bundle E over X and a holomorphic section \phi of E. There is a concept of stability for pairs which depends on a real parameter \tau. Here we prove that the third cohomology groups of the moduli spaces of \tau-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank at least two and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H^1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.