Prym varieties of spectral covers

Hausel, Tamás¹ ; Pauly, Christian²

¹ Section de Mathématiques, École Polytechnique Fédéral de Lausanne, Section 8, CH-1015 Lausanne, Switzerland
² Laboratoire de Mathématiques J.A. Dieudonné, UMR no 7351 CNRS UNSA, Université de Nice Sophia-Antipolis, 06108 Nice Cedex 02, France

Geometry & topology, Tome 16 (2012) no. 3, pp. 1609-1638.

Résumé

Given a possibly reducible and non-reduced spectral cover $\pi :X\to C$ over a smooth projective complex curve $C$ we determine the group of connected components of the Prym variety $Prym\left(X∕C\right)$. As an immediate application we show that the finite group of $n$–torsion points of the Jacobian of $C$ acts trivially on the cohomology of the twisted ${SL}_n$–Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder–Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted ${SL}_n$ stable bundle moduli space.

Détail

DOI : 10.2140/gt.2012.16.1609

Keywords: Prym varieties, Hitchin fibration, Higgs bundles, vector bundles on curves

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