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Torelli theorem for the moduli spaces of pairs

Published online by Cambridge University Press:  01 May 2009

VICENTE MUÑOZ
VICENTE MUÑOZ*
Affiliation:
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, Serrano 113 bis, 28006 Madrid, Spain. and Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain. e-mail: [email protected]
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Abstract

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. A pair (E, φ) over X consists of an algebraic vector bundle E over X and a section φ ∈ H0(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 n ≥ 2 are polarised pure Hodge structures, and they are isomorphic to H1(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 H1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 675 - 693
Copyright
Copyright © Cambridge Philosophical Society 2008

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