Torelli theorem for the moduli spaces of pairs

Abstract

Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,ϕ) over X consists of an algebraic vector bundle E over X and a holomorphic section ϕof E. There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-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.

22 pages

References in corpus (2)

• A universal construction for moduli spaces of decorated vector bundles over curves
• Moduli spaces of holomorphic triples over compact Riemann surfaces

Cited by in corpus (7)

• On moduli spaces of Hitchin pairs
• Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles
• Quadric bundles and their moduli spaces
• Torelli theorem for the moduli spaces of rank 2 quadratic pairs
• Hitchin Pairs for non-compact real Lie groups
• Hodge structures of the moduli spaces of pairs
• Torelli theorem for moduli spaces of SL(r,C)-connections on a compact Riemann surface