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Torelli theorem for the moduli spaces of pairs

Muñoz, Vicente (2009) Torelli theorem for the moduli spaces of pairs. Mathematical Proceedings of the Cambridge Philosophical Society, 146 (3). pp. 675-693. ISSN 0305-0041

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Abstract

Let X be a smooth projective curve of genus g >= 2 over C. A pair (E, phi) over X consists of an algebraic vector bundle E over X and a section phi is an element of H(0)(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 n >= 2 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 n >= 2 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.


Item Type:Article
Uncontrolled Keywords:Ppolystable pair; Semistable vector bundles; Semistable triple; Moduli space; Smooth projective curve; Torelli theorem; Hodge structure
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:17030
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